Intelligent Road Transport System Literature - Myassignmenthelp.Com

Question: Discuss about theIntelligent Road Transport System Literature. Answer: Arterial and Freeway management systems The traffic jamming is becoming one of the serious road concerns now a day, worldwide. The main concern is the rising population, high vehicle count, low infrastructure and to worsen the conditions, a bad traffic management system. To mitigate this concern, Arterial and Freeway management systems can be adopted (Demissie, Almeida Correia and Bento 2013). This particular system comprises of installations, which are used for the purpose if controlling and monitoring the transport systems. In case of the freeway management systems, around 2 gigabytes of data are processed in real time, which would be used for the collection and storage of data related to the speed records, the traffic flow, the traffic pattern and many more. This system can be accompanied by the collaborative processing of the emerging technologies and hardware systems like the Variable Message Sign (VMS), ramp meter, close circuit television system (CCTV) and traffic signal control system (S?adkowski and Pamu?a 2015). References Demissie, M.G., de Almeida Correia, G.H. and Bento, C., 2013. Intelligent road traffic status detection system through cellular networks handover information: An exploratory study.Transportation research part C: emerging technologies,32, pp.76-88. S?adkowski, A. and Pamu?a, W. eds., 2015.Intelligent Transportation SystemsProblems and Perspectives(Vol. 32). Springer.

Research Paper Outlines Can Be Fun for Everyone

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Information About the Chinas Next Economy Paper Topics Quizlet

Information About the China's Next Economy Paper Topics QuizletThe next economy quizlet will be on the Chinese economy, or why should you study China's next economy. You will also find this quilt very interesting and most of the time you can learn a lot from it.The first thing you need to understand about the economy quizlet is that it is based on a multi-level challenge. What the quizlet does is it will require you to answer some complex questions about the Chinese economy.In this case it is better to say that it is a test or a series of tests for one to know how well they can answer questions regarding that. Now, you may be wondering what are these questions and which subjects will be covered in the quizzes. Well, you will find a series of question in the economy quiet, but the test is made up of a series of areas.This can be said as two levels, each level being really difficult. The first level will be level one, where the quizlet will ask you questions about four points. The ques tions you will have to answer will be about the following: taxes, spending, economic cycle, and central government.On the second level, the quizlet will ask for two-point questions about certain topics. The questions will be asked to be about market allocation, information supply, stimulus packages, debt relief, foreign investment, and other issues.Lastly, there is one question which is added at the end of the quilt, which will ask for your honest opinion. This question will ask you to take a survey. In this case, this question will also take into account the time you spend in doing the exam.After reading the paper topics quiet, you can find a good way to study this question correctly. In fact, all of the above topics are indeed part of the paper topics quiet. Therefore, studying the paper topics quizzes will help you in understanding China's economy.This means that you can gain a lot of information when studying China's economy. It is important to note that the quiet can be difficu lt and you have to put your best effort to study it. You should know that a good way to study the quizlet is by studying it with a group of people.

A History of Cause and Effect Essay Topics Tumblr Refuted

A History of Cause and Effect Essay Topics Tumblr Refuted What Is So Fascinating About Cause and Effect Essay Topics Tumblr? Kibin editors are content to help review your paper. The group of qualified essay writers is always prepared to aid you with that in no moment! Though some may see them as pretty complicated, they are a few of the most straightforward article ideas you are ever going to run into. This informative article is a short description on sample topics that could guide you when you're rushing for writing your next academic essay. Great essay topics might be found online, but the majority of them are pretty old and not intriguing. Don't forget to provide a concise background of this issue and write your thesis statement. Finally, you've got to reassess the paper on multiple occasions. Within this paper, you may want to examine both the beneficial and negative outcomes. Not merely it will permit you learn more regarding the outline, but help you to distinguish between positive and negative papers. Cause and Effect Essay Topics Tumblr - Overview Cause and effect is a typical way of organizing and discussing ideas. The negative effect of boxing. The psychological consequences of qualified sport. The explanations for playing dangerous sports and its consequences on health. The Cause and Effect Essay Topics Tumblr Cover Up The very first thing about subtitles is they aren't synonymous with closed captions. While using dashes is a particular type of explicit spoiler, subtitles may also spoil a minute implicitly. Regardless of what way you decide on, you've got to compose reasonable topic sentences to each paragraph. Then provide a thorough explanation utilizing appropriate transitional words. How to Get Started with Cause and Effect Essay Topics Tumblr? You won't need to be concerned about anything and are going to be able to center on your own personal matters or other assignments! Well, the majority of these topics are for the most part taught within a course setting. There are a lot of forms of essays there in the world you can merely lose your head. This sort of essay demands the student to employ creativity in addition to a high level of analytical skills. The assignment is usually given to assess along with help students develop the capability to assess situations, determine the reason and earn a deeper knowledge of a function. Cause and effect is a huge pick for those who wish to better their skills not just in writing but logical thinking also. The very first unfavorable effect technology has had on my life is the period of time it consumes. The factors for wearing protective gear and the advantages of it. Facts, Fiction and Cause and Effect Essay Topics Tumblr There are various sorts of cause and effect that someone may take under consideration. Take your time to thoroughly examine our cause and effect topics list till you locate a prompt that you're excited to write about. There are many regions to explore while trying to find suitable cause and effect topics. More often you will need to select your own cause and effect essay topic. For example, the essay may just concentrate on the causes of a certain actions. At precisely the same time, detecting cause and effect relationships isn't that easy in regards to the selection of a very good cause and effect essay topic. If you are searching for cause and effect essay examples, here's a great one below. You can begin by describing effects. Our professional writers are pleased to help you with any cause and effect project that you will need to get written. Some are difficult others straightforward. In the event you are in need of a cause and effect essay help don't hesitate to speak to our support team for extra support. Describe at least two distinct theories about exactly what causes someone to dream.

An Introduction to Short Story Analysis Essays

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Importance Impact of Ethical Communication free essay sample

Abstract Communication is constant, especially in the information age. Savvy professionals know how to communicate quickly, effectively and ethically. The term ethical communication has different meanings depending on the context. A shampoo advertiser and a sports team spokesperson may have dissimilar views on what constitutes as ethical communication. Some communication guidelines are only applicable to certain situations, while others could be understood as ethical in one situation and unethical in another. Every aspect of ethical communication should be considered within the boundaries of the issue at hand. Introduction The Business Dictionary defines ethical standards as follows: Principles that when followed, promote values such as trust, good behavior, fairness, and/or kindness. There is not one consistent set of standards that all companies follow, but each company has the right to develop the standards that are meaningful for their organization. Ethical standards are not always easily enforceable, as they are frequently vaguely defined and somewhat open to interpretation (Men and women should be treated equally, or Treat the customer with respect and kindness. We will write a custom essay sample on Importance Impact of Ethical Communication or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page ). Others can be more specific, such as Do not share the customers private information with anyone outside of the company. Ethical Communication in Business Every business is dependent on effective ethical communication. Its what makes new policy in government, raises money for nonprofits and strengthens a business. Business communication occurs any time a message is given or received, whether its verbal or nonverbal, between two businesses, a business and its employees or a business and the public. The messages sent and received by a business need to follow ethical norms that dont offend or make individuals feel uncomfortable. Significance Ethical business communication holds great significance on three main fronts: business to business, business to employees and business to the public. An example of the first type is between a business and its suppliers or distributors. The second is organizational communication within the business itself, how the leadership communicates with the employees. Lastly, communication with the public is how the business develops its public image. Maintaining high ethical standards on each front is essential to success in business. Function Ethical business communications primary function is to send and receive messages in a neutral, non-offensive manner. Ethical effective communication skills in business strengthen its corporate culture, resulting in a more attractive bottom line. When communication does not adhere to ethical standards, the consequences can include unhappy employees, a poor public image and a decrease in the bottom line. Ethical business communication is intended to care for the emotional and perceptive needs of its employees and customers. Misconceptions A common misconception concerning ethical communication in business is that most unethical communication is obvious and overt. True unethical communication is based on perception. If a person or people group perceive or interpret certain actions or words to be discriminatory or offensive, the communication can be considered unethical. The same is true with interpersonal interactions between employees. If a particular form of communication or gesture is offensive to another employee, it can be categorized as unethical. Purpose The purpose of ethical communication in business is to protect, respect and maintain a good public image. The communication in any business is for the purpose of maintaining order and the proper image with its employees and society. For example, if an accusation was to arise against a particular company, the public relations representative will arrange a press conference to verbally address the issue at hand. The company may also change a policy pertaining to the issue that non-verbally communicates the same message. Consequences There are many consequences to unethical business communication. A business may have an outstanding product or service, but if it doesnt communicate well with its customers, they will not be satisfied, and this can weaken the business/customer relationship. Everything in business rises and falls on communication. When ethical communication is lacking, moral, corporate image and motivation will lack as well. Each of these elements affects the spirit of the employees, which in turn will reflect to the customer and result in a decrease in revenue. Ethical behavior is a companywide concern, of course; but because communication efforts are the public face of a company, they are subjected to particularly rigorous scrutiny from regulators, legislators, investors, consumer groups, environmental groups, labor organizations, and anyone else affected by business activities. Ethical communication includes all relevant information, is true in every sense, and is not deceptive in any way. In contrast, unethical communication can include falsehoods and misleading information (or can withhold important information). Some examples of unethical communication include: Plagiarism: Plagiarism is presenting someone else’s words or other creative product as your own. Note that plagiarism can also be illegal if it violates a copyright, which is a form of legal protection for the expression of creative ideas. Selective misquoting: Deliberately omitting damaging or unflattering comments to paint a better (but untruthful) picture of you or your company. Misrepresenting numbers: Increasing or decreasing numbers, exaggerating, altering statistics, or omitting numerical data. Distorting visuals: Making a product look bigger or changing the scale of graphs and charts to exaggerate or conceal differences. In contrast, an ethical message is accurate and sincere. It avoids language and images that manipulate, discriminate, or exaggerate. On the surface, such ethical practices appear fairly easy to recognize, but deciding what is ethical can be a considerable challenge in complex business situations. Distinguishing Ethical Dilemmas from Ethical Lapses Every company has responsibilities to its stakeholders, and those various groups often have competing interests. An ethical dilemma involves choosing among alternatives that aren’t clear-cut. Perhaps two conflicting alternatives are both ethical and valid, or perhaps the alternatives lie somewhere in the gray area between clearly right and clearly wrong. Suppose you are the chief executive of a company whose sales are declining and you might be forced to reduce costs by laying off 100 employees. You’ve decided to wait two months before making this tough decision. Here’s your dilemma: Do you tell the workforce now that several hundred jobs could disappear in the near future? Telling them now would give people more time to look for new jobs and adjust their finances—clearly a good thing. However, if you tell them now, vital employees nervous about their future could jump ship, which could drive sales down even more—clearly not a good thing. And what if you tell them now and many people leave but then sales improve enough in the next two months that you can avoid the layoffs? You will have unnecessarily disrupted many careers and families. Situations such as these often have no clear answer. In contrast, an ethical lapse is a clearly unethical (and frequently illegal) choice. In 2004, several insurance companies were accused of misleading military personnel at Fort Benning in Georgia, Camp Pendleton in California, and other bases around the country. Many of these young men and women thought they were signing up for savings programs when in fact they were buying extremely expensive and frequently unnecessary life insurance policies. The policies were often sold during mandatory financial training sessions for the soldiers, who were given no time to read the documents they signed. After the situation was brought to national attention by the New York Times and other news media, at least two of the companies involved, Madison National Life Insurance Company and American Amicable Life Insurance, began issuing full refunds. Ensuring Ethical Communication Ensuring ethical business communication requires three elements: ethical individuals, ethical company leadership, and the appropriate policies and structures to support employees’ efforts to make ethical choices. 39 Moreover, these three elements need to work in harmony. If employees see company executives making unethical decisions and flouting company guidelines, they might conclude that the guidelines are meaningless and emulate their bosses’ unethical behavior. Employers have a responsibility to establish clear guidelines for ethical behavior, including business communication. In a recent global survey by the International Association of Business Communicators, 70 percent of communication professionals said their companies clearly define what is considered ethical and unethical behavior. On a somewhat less positive note, slightly fewer than half said their companies encourage open discussion of ethical issues and dilemmas. Many companies establish an explicit ethics policy by using a written code of ethics to help employees determine what is acceptable. A code is often part of a larger program of employee training and communication channels that allow employees to ask questions and report instances of questionable ethics. For example, United Technologies, a large aerospace and defense company based in Hartford, Connecticut, offers employees, customers, and suppliers a confidential way to report suspected fraud and other ethical concerns. The people who share their concerns through the program then receive a written response that explains how the situation was resolved. 1 To ensure ongoing compliance with their codes of ethics, many companies also conduct ethics audits to monitor ethical progress and to point out any weaknesses that need to be addressed. Principles of Ethical Communication The Business Dictionary defines ethical standards as Principles that when followed, promote values such as trust, good behavior, fairness, and/or kindness. In order to establish good communication with people of other cultures, it is essential to understand their ethical framework. In rder to learn ethical intercultural communication, you must expect people of other cultures to think differently, be willing to learn culturally appropriate behavior and (at least to some extent) practice what their cultures consider ethical. Expect Differences Ethical principles are not the same across cultures. Rather, ethics are culturally informed. The right thing to do is not just instinctive in humans. Many aspects of what is good are taught (consciously and subconsciously) by a persons culture. So, if you want to establish ethical intercultural communication with people of another background, prepare yourself to see the world differently. Do not expect that what seems good to you will also seem good to them; understand that they may view some things as bad that you view as fine or good. For example, while American culture teaches that individuality is a good thing and that standing on your own two feet is a position you should strive for, many other cultures value the group more than the individual. For example, in most African cultures, being part of a strong family support system is considered much more important and valuable than standing alone as an individual. Exemplify the Universal Although different cultures will have varying expectations and standards as to what is ethical, there are some ethical standards that are universal. So, by striving to abide by ethical standards that are universally received, you can take the first step in communicating and connecting well with people of another culture. According to William Howell in his Ethics of Intercultural Communication, Two principles that are universal are that no action is ethical if it harms persons, and the action that benefits persons accumulates ethical quality. Act in such a way that you do not intentionally bring harm to anyone, and always keep others best interests in mind. Learn their Culture To really communicate well interculturally, you must ask the question: What makes a good person in your culture? Talk with people in the target culture to discover the traits of an ethical person. What attitudes and actions does a good person possess? Does a good person set aside his personal work to take care of his parents when they are elderly? Does a good person control his anger at all times? Does a good person practice abstinence in certain areas? You will find, as you look into someone elses culture, that the things that make a good person in your culture are not the same things that comprise a good person in every culture. Empathize Through Action As you learn the ways of another culture, the best way to establish good intercultural communication is to act in a way that is considered ethical in that culture. Do and say the things that will express that you have the best interest of those around you in mind. Enjoy the food people prepare for you. If there are certain respectful gestures associated with greeting people older or more prestigious than yourself (or everyone), use them. Learn at least enough of the language to greet people and ask how they are doing in their native tongue. Wear clothing that is culturally appropriate. Respect family organization and methods of doing education and business. On every level of life and society, share in the way people think and act as much as you are able to. This willingness to adopt the standards of another culture is the best way to establish ethical and intercultural communication. Goals of Ethical Business Communications The purpose of business is to make money. Behaving ethically serves that purpose. People prefer doing business with ethical companies, companies they can trust, so in the long run the ethical company benefits from its behavior. This means that the goal of ethical business communication is to build the trust and credibility of the company. The International Association of Business Communicators maintains that companies that the practice of ethical business communication also increases a team feeling among employees and boosts employee morale. To accomplish these goals, corporate communication must strive to attain certain specific ethical goals. Honesty It is to a companys benefit to be honest. Honesty is the basis of trust. If others feel that they can believe what a company says, they will trust it. Other factors being equal, people prefer doing business with a company they can trust. Honesty means saying what you believe to be true, but it also means distinguishing fact from opinion. It is easy to disguise opinion as fact. Some television news commentators do it every day, and their credibility suffers for it. They may be considered entertaining, but what they say is taken with a grain of salt. Consultant Michelle Howe advises any company that wants to be trusted to clearly label opinion as such, and to present what it has to say in an unbiased manner. Clarity Distinguishing fact from opinion is part of a larger goal of being clear and easy to understand. Ethical business communication calls for being clearly understood. It means that the company is not seen as attempting to obfuscate or confuse the public and other companies with whom it does business. Timeliness of communication can also help. Within the company, acknowledging problems and keeping relevant people informed with clear and direct communications helps dampen the rumor mill and maintains better employee morale. Commitment In the context of business communications, commitment means allocating the necessary time and resources to discussing issues fully. Communication needs to be thorough, for only when time and resources, such as feedback forms, are dedicated to discussing issues is there a chance for everyone in the organization to have their voice heard. Acknowledging Sources Few things create as much tension as when someone presents another persons ideas as his own. Employees want credit for their work, so failure to acknowledge them is not only unethical but also bad for morale. Some people believe that concerns about plagiarism are only important in academic ettings, but anytime someone is caught borrowing someone elses ideas without proper acknowledgment, trustworthiness takes a nosedive. Most people realize its important to use quotations when citing direct statements from others, but its also good practice and sound business to acknowledge ideas that are not your own. Openness to Other Views Openness is one of the key pillars of ethical communication. In communication, openness means being open to diverse ideas and opinions, as well as being ready to offer your own opinions even if you do not think they will be popular. A business environment where people are not free to play the devils advocate and say unpopular opinions is not an ethical one, because intolerance of divergent opinions means intolerance of differences and free flow of information is essential to both the publics and the organizations long-term well-being. Taking Care with Confidential Information Confidential information is a special class of information that requires special attention. The North Carolina State University business department emphasizes the importance of the ethical business practice of protecting confidential information while complying with public disclosure laws. Any use of confidential information for personal gain is also clearly unethical. IABC Code of Ethics for Professional Communicators Preface Because hundreds of thousands of business communicators worldwide engage in activities that affect the lives of millions of people, and because this power carries with it significant social responsibilities, the International Association of Business Communicators developed the Code of Ethics for Professional Communicators. The Code is based on three different yet interrelated principles of professional communication that apply throughout the world. These principles assume that just societies are governed by a profound respect for human rights and the rule of law; that ethics, the criteria for determining what is right and wrong, can be agreed upon by members of an organization; and, that understanding matters of taste requires sensitivity to cultural norms. These principles are essential: †¢Professional communication is legal. †¢Professional communication is ethical. †¢Professional communication is in good taste. Recognizing these principles, members of IABC will: Engage in communication that is not only legal but also ethical and sensitive to cultural values and beliefs; †¢Engage in truthful, accurate and fair communication that facilitates respect and mutual understanding; †¢adhere to the following articles of the IABC Code of Ethics for Professional Communicators. Because conditions in the world are constantly changing, members of IABC will work to improve their individual competence and to increase the body of knowledge in the field with research and education. Articles 1. Professional communicators uphold the credibility and dignity of their profession by practicing honest, candid and timely communication and by fostering the free flow of essential information in accord with the public interest. 2. Professional communicators disseminate accurate information and promptly correct any erroneous communication for which they may be responsible. 3. Professional communicators understand and support the principles of free speech, freedom of assembly, and access to an open marketplace of ideas and act accordingly. 4. Professional communicators are sensitive to cultural values and beliefs and engage in fair and balanced communication activities that foster and encourage mutual understanding. 5. Professional communicators refrain from taking part in any undertaking which the communicator considers to be unethical. 6. Professional communicators obey laws and public policies governing their professional activities and are sensitive to the spirit of all laws and regulations and, should any law or public policy be violated, for whatever reason, act promptly to correct the situation. 7. Professional communicators give credit for unique expressions borrowed from others and identify the sources and purposes of all information disseminated to the public. 8. Professional communicators protect confidential information and, at the same time, comply with all legal requirements for the disclosure of information affecting the welfare of others. 9. Professional communicators do not use confidential information gained as a result of professional activities for personal benefit and do not represent conflicting or competing interests without written consent of those involved. 0. Professional communicators do not accept undisclosed gifts or payments for professional services from anyone other than a client or employer. 11. Professional communicators do not guarantee results that are beyond the power of the practitioner to deliver. 12. Professional communicators are honest not only with others but also, and most importantly, with themselves as individuals; for a professional commu nicator seeks the truth and speaks that truth first to the self. Enforcement and Communication of the IABC Code of Ethics IABC fosters compliance with its Code by engaging in global communication campaigns rather than through negative sanctions. However, in keeping with the sixth article of the IABC Code, members of IABC who are found guilty by an appropriate governmental agency or judicial body of violating laws and public policies governing their professional activities may have their membership terminated by the IABC executive board following procedures set forth in the associations bylaws. IABC encourages the widest possible communication about its Code. The IABC Code of Ethics for Professional Communicators is published in several languages and is freely available to all: Permission is hereby granted to any individual or organization wishing to copy and incorporate all or part of the IABC Code into personal and corporate codes, with the understanding that appropriate credit be given to IABC in any publication of such codes. The IABC Code is published on the association’s web site. The association’s bimonthly magazine, Communication World, publishes periodic articles dealing with ethical issues. At least one session at the association’s annual conference is devoted to ethics. The international headquarters of IABC, through its professional development activities, encourages and supports efforts by IABC student chapters, professional chapters, and regions to conduct meetings and workshops devoted to the topic of ethics and the IABC Code. New and renewing members of IABC sign the following statement as part of their application: â€Å"I have reviewed and understand the IABC Code of Ethics for Professional Communicators. As a service to communicators worldwide, inquiries about ethics and questions or comments about the IABC Code may be addressed to members of the IABC Ethics Committee. The IABC Ethics Committee is composed of at least three accredited members of IABC who serve staggered three-year terms. Other IABC members may serve on the committee with the approval of the IABC executive committee. The functions of the Ethics Committee are to assist with professional d evelopment activities dealing with ethics and to offer advice and assistance to individual communicators regarding specific ethical situations.

Complete Guide to Integers on SAT Math (Advanced)

Complete Guide to Integers on SAT Math (Advanced) SAT / ACT Prep Online Guides and Tips Integer questions are some of the most common on the SAT, so understanding what integers are and how they operate will be crucial for solving many SAT math questions. Knowing your integers can make the difference between a score you’re proud of and one that needs improvement. In our basic guide to integers on the SAT (which you should review before you continue with this one), we covered what integers are and how they are manipulated to get even or odd, positive or negative results. In this guide, we will cover the more advanced integer concepts you’ll need to know for the SAT. This will be your complete guide to advanced SAT integers, including consecutive numbers, primes, absolute values, remainders, exponents, and roots- what they mean, as well as how to handle the more difficult integer questions the SAT can throw at you. Typical Integer Questions on the SAT Because integer questions cover so many different kinds of topics, there is no â€Å"typical† integer question. We have, however, provided you with several real SAT math examples to show you some of the many different kinds of integer questions the SAT may throw at you. Over all, you will be able to tell that a question requires knowledge and understanding of integers when: #1: The question specifically mentions integers (or consecutive integers). Now this may be a word problem or even a geometry problem, but you will know that your answer must be in whole numbers (integers) when the question asks for one or more integers. If $j$, $k$, and $n$ are consecutive integers such that $0jkn$ and the units (ones) digit of the product $jn$ is 9, what is the units digit of $k$? A. 0B. 1C. 2D. 3E. 4 (We will go through the process of solving this question later in the guide) #2: The question deals with prime numbers. A prime number is a specific kind of integer, which we will discuss in a minute. For now, know that any mention of prime numbers means it is an integer question. What is the product of the smallest prime number that is greater than 50 and the greatest prime number that is less than 50? (We will go through the process of solving this question later in the guide) #3: The question involves an absolute value equation (with integers) Anything that is an absolute value will be bracketed with absolute value signs which look like this:| | For example: $|-210|$ or $|x + 2|$ $|10 - k| = 3$ $|k - 5| = 8$ What is a value for k that fulfills both equations above? (We will go through how to solve this problem in the section on absolute values below) Note: there are several different kinds of absolute value problems. About half of the absolute value questions you come across will involve the use of inequalities (represented by $$ or $$). If you are unfamiliar with inequalities, check out our guide to inequalities. The other types of absolute value problems on the SAT will either involve a number line or a written equation. The absolute value questions involving number lines almost always use fraction or decimal values. For information on fractions and decimals, look to our guide to SAT fractions. We will be covering only written absolute value equations (with integers) in this guide. #4: The question uses perfect squares or asks you to reduce a root value A root question will always involve the root sign: $√$ $√81$, $^3√8$ You may be asked to reduce a root, or to find the square root of a perfect square (a number that is the square of an integer). You may also need to multiply two or more roots together. We will go through these definitions as well as how all of these processes are done in the section on roots. (Note: A root question with perfect squares may involve fractions. For more information on this concept, look to our guide on fractions and ratios.) #5: The question involves multiplying or dividing bases and exponents Exponents will always be a number that is positioned higher than the main (base) number: $2^7$, $(x^2)^4$ You may be asked to find the values of exponents or find the new expression once you have multiplied or divided terms with exponents. We will go through all of these questions and topics throughout this guide in the order of greatest prevalence on the SAT. We promise that integers are a whole lot less mysterious than...whatever these things are. Exponents Exponent questions will appear on every single SAT, and you will likely see an exponent question at least twice per test. An exponent indicates how many times a number (called a â€Å"base†) must be multiplied by itself. So $4^2$ is the same thing as saying $4 * 4$. And $4^5$ is the same thing as saying $4 * 4 * 4 * 4 * 4$. Here, 4 is the base and 2 and 5 are the exponents. A number (base) to a negative exponent is the same thing as saying 1 divided by the base to the positive exponent. For example, $2^{-3}$ becomes $1/2^3$ = $1/8$ If $x^{-1}h=1$, what does $h$ equal in terms of $x$? A. $-x$B. $1/x$C. $1/{x^2}$D. $x$E. $x^2$ Because $x^{-1}$ is a base taken to a negative exponent, we know we must re-write this as 1 divided by the base to the positive exponent. $x^{-1}$ = $1/{x^1}$ Now we have: $1/{x^1} * h$ Which is the same thing as saying: ${1h}/x^1$ = $h/x$ And we know that this equation is set equal to 1. So: $h/x = 1$ If you are familiar with fractions, then you will know that any number over itself equals 1. Therefore, $h$ and $x$ must be equal. So our final answer is D, $h = x$ But negative exponents are just the first step to understanding the many different types of SAT exponents. You will also need to know several other ways in which exponents behave with one another. Below are the main exponent rules that will be helpful for you to know for the SAT. Exponent Formulas: Multiplying Numbers with Exponents: $x^a * x^b = x^[a + b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $2^4 * 2^6$, you have: $(2 * 2 * 2 * 2) * (2 * 2 * 2 * 2 * 2 * 2)$ If you count them, this give you 2 multiplied by itself 10 times, or $2^10$. So $2^4 * 2^6$ = $2^[4 + 6]$ = $2^10$. If $7^n*7^3=7^12$, what is the value of $n$? A. 2B. 4C. 9D. 15E. 36 We know that multiplying numbers with the same base and exponents means that we must add those exponents. So our equation would look like: $7^n * 7^3 = 7^12$ $n + 3 = 12$ $n = 9$ So our final answer is C, 9. $x^a * y^a = (xy)^a$ (Note: the exponents must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $2^4 * 3^4$, you have: $(2 * 2 * 2 * 2) * (3 * 3 * 3 * 3)$ = $(2 * 3) * (2 * 3) * (2 * 3) * (2 * 3)$ So you have $(2 * 3)^4$, or $6^4$ Dividing Exponents: ${x^a}/{x^b} = x^[a-b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. ${2^6}/{2^2}$ can also be written as: ${(2 * 2 * 2 * 2 * 2 * 2)}/{(2 * 2)}$ If you cancel out your bottom 2s, you’re left with $(2 * 2 * 2 * 2)$, or $2^4$ So ${2^6}/{2^2}$ = $2^[6-2]$ = $2^4$ If $x$ and $y$ are positive integers, which of the following is equivalent to $(2x)^{3y}-(2x)^y$? A. $(2x)^{2y}$B. $2^y(x^3-x^y)$C. $(2x)^y[(2x)^{2y}-1]$D. $(2x)^y(4x^y-1)$E. $(2x)^y[(2x)^3-1]$ In this problem, you must distribute out a common element- the $(2x)^y$- by dividing it from both pieces of the expression. This means that you must divide both $(2x)^{3y}$ and $(2x)^y$ by $(2x)^y$. Let's start with the first: ${(2x)^{3y}}/{(2x)^y}$ Because this is a division problem that involves exponents with the same base, we say: ${(2x)^{3y}}/{(2x)^y} = (2x)^[3y - y]$ So we are left with: $(2x)^{2y}$ Now, for the second part of our equation, we have: ${(2x)^y}/{(2x)^y}$ Again, we are dividing exponents that have the same base. So by the same process, we would say: ${(2x)^y}/{(2x)^y} = (2x)^[y - y] = (2x)^0 = 1$ (Why 1? Because, as you'll see below, anything raised to the power of 0 = 1) So our final answer looks like: ${(2x)^y}{((2x)^{2y} - 1)}$ Which means our final answer is C. Taking Exponents to Exponents: $(x^a)^b = x^[a * b]$ Why is this true? Think about it using real numbers. $(2^3)^4$ can also be written as: $(2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2)$ If you count them, 2 is being multiplied by itself 12 times. So $(2^3)^4 = 2^[3 * 4] = 2^12$ $(x^y)^6 = x^12$, what is the value of $y$? A. 2B. 4C. 6D. 10E. 12 Because exponents taken to exponents are multiplied together, our problem would look like: $y * 6 = 12$ $y = 2$ So our final answer is A, 2. Distributing Exponents: $(x/y)^a = {x^a}/{y^a}$ Why is this true? Think about it using real numbers. $(2/4)^3$ can be written as: $(2/4) * (2/4) * (2/4)$ $8/64 = 1/8$ You could also say $2^3/4^3$ = $8/64$ = $1/8$ $(xy)^z = x^z * y^z$ If you are taking a modified base to the power of an exponent, you must distribute that exponent across both the modifier and the base. $(3x)^3$ = $3^3 * x^3$ (Note on distributing exponents: you may only distribute exponents with multiplication or division- exponents do not distribute over addition or subtraction. $(x + y)^a$ is NOT $x^a + y^a$, for example) Special Exponents: For the SAT you should know what happens when you have an exponent of 0: $x^0=1$ where $x$ is any number except 0 (Why any number but 0? Well 0 to any power other than 0 is 0, because $0x = 0$. And any other number to the power of 0 is 1. This makes $0^0$ undefined, as it could be both 0 and 1 according to these guidelines.) Solving an Exponent Question: Always remember that you can test out exponent rules with real numbers in the same way that we did above. If you are presented with $(x^2)^3$ and don’t know whether you are supposed to add or multiply your exponents, replace your x with a real number! $(2^2)^3 = (4)^3 = 64$ Now check if you are supposed to add or multiply your exponents. $2^[2+3] = 2^5 = 32$ $2^[2 * 3] = 2^6 = 64$ So you know you’re supposed to multiply when exponents are taken to another exponent. This also works if you are given something enormous, like $(x^23)^4$. You don’t have to test it out with $2^23$! Just use smaller numbers like we did above to figure out the rules of exponents. Then, apply your newfound knowledge to the larger problem. And the philosophical debate continues. Roots Root questions are common on the SAT, and you should expect to see at least one during your test. Roots are technically fractional exponents. You are likely most familiar with square roots, however, so you may have never heard a root expressed in terms of exponents before. A square root asks the question: "What number needs to be multiplied by itself one time in order to equal the number under the root sign?" So $√36 = 6$ because 6 must be multiplied by itself one time to equal 36. In other words, $6^2 = 36$ Another way to write $√36$ is to say $^2√36$. The 2 at the top of the root sign indicates how many numbers (2 numbers, both the same) are being multiplied together to become 36. (Note: you do not expressly need the 2 at the top of the root sign- a root without an indicator is automatically a square root.) So $^3√27 = 3$ because three numbers, all of which are the same ($3 * 3 * 3$), multiplied together equals 27. Or $3^3 = 27$. Fractional Exponents If you have a number to a fractional exponent, it is just another way of asking you for a root. So $16^{1/2} = ^2√16$ To turn a fractional exponent into a root, the denominator becomes the value to which you take the root. But what if you have a number other than 1 in the numerator? $16^{2/3} = ^3√16^2$ The denominator becomes the value to which you take the root, and the numerator becomes the exponent to which you take the number under the root sign. Distributing Roots $√xy = √x * √y$ Just like with exponents, roots can be separated out. So $√20$ = $√2 * √10$ or $√4 * √5$ $√x * √y = √xy$ Because they can be separated, roots can also come together. So $√2 * √10$ = $√20$ Reducing Roots It is common to encounter a problem with a mixed root, where you have an integer multiplied by a root (like $3√2$). Here, $3√2$ is reduced to its simplest form, but let's say you had something like this instead: $2√12$ Now $2√12$ is NOT as reduced as it can be. In order to reduce it, we must find out if there are any perfect squares that factor into 12. If there are, then we can take them out from under the root sign. (Note: if there is more than one perfect square that can factor into your number under the root sign, use the largest one.) 12 has several factor pairs. These are: $1 * 12$ $2 * 6$ $3 * 4$ Well 4 is a perfect square because $2 * 2 = 4$. That means that $√4 = 2$. This means that we can take 4 out from under the root sign. Why? Because we know that $√xy = √x * √y$. So $√12 = √4 * √3$. And $√4 = 2$. So 4 can come out from under the root sign and be replaced by 2 instead. $√3$ is as reduced as we can make it, since it is a prime number. We are left with $2√3$ as the most reduced form of $√12$ (Note: you can test to see if this is true on most calculators. $√12 = 3.4641$ and $2 *√3 = 2 * 1.732 = 3.4641$. The two expressions are identical.) Now to finish the problem, we must multiply our reduced form of $√12$ by 2. Why? Because our original expression was $2√12$. $2 * 2√3 = 4√3$ So $2√12$ in its most reduced form is $4√3$ Remainders Questions involving remainders generally show up at least once or twice on any given SAT. A remainder is the amount left over when two numbers do not divide evenly. If you divide 12 by 4, you will not have any remainder (your remainder will be zero). But if you divide 13 by 4, you will have a remainder of 1, because there is 1 left over. You can think of the division as $13/4 = 3{1/4}$. That extra 1 is left over. Most of you probably haven’t worked with integer remainders since elementary school, as most higher level math classes and questions use decimals to express the remaining amount after a division (for the above example, $13/4 = 3 \remainder 1$ or $3.25$). But for some situations, decimals simply do not apply. Joanne’s hens laid a total of 33 eggs. She puts them into cartons that fit 6 eggs each. How many eggs will she have left that do NOT make a full carton of eggs? $33/6 = 5 \remainder 3$. So Joanne can make 5 full baskets with 3 eggs left over. Some remainder questions may seem incredibly obscure, but they are all quite basic once you understand what is being asked of you. Which of the following answers could be the remainders, in order, when five positive consecutive integers are divided by 4? A. 0, 1, 2, 3, 4B. 2, 3, 0, 1, 2C. 0, 1, 2, 0, 1D. 2, 3, 0, 3, 2E. 2, 3, 4, 3, 2 This question may seem complicated at first, so let’s break it down into pieces. The question is asking us to find the list of remainders when positive consecutive integers are divided by 4. This means we are NOT looking for the answer plus remainders- we are just trying to find the remainders by themselves. We will discuss consecutive integers below in the guide, but for now understand that "positive consecutive integers" means positive integers in a row along a number line. So positive consecutive integers increase by 1 continuously. , 12, 13, 14, 15, etc. are an example of positive consecutive integers. We also know that any number divided by 4 can have a maximum remainder of 3. Why? Because if any number could be divided by 4 with a remainder of 4 left over, it means it could be divided by 4 one more time! For example, $16/4 = 4 \remainder 0$ because 4 goes into 16 exactly 4 times. (It is NOT $3 \remainder 4$.) So that automatically lets us get rid of answer choices A and E, as those options both include a 4 for a remainder. Now we also know that, when positive consecutive integers are divided by any number, the remainders increase by 1 until they hit their highest remainder possible. When that happens, the next integer remainder resets to 0. This is because our smaller number has gone into the larger number an even number of times (which means there is no remainder). For example, $10/4 = 2 \remainder 2$, $/4 = 2 \remainder 3$, $12/4 = 3 \remainder 0$, and $13/4 = 3 \remainder 1$ Once the highest remainder value is achieved (n - 1, which in this case is 3), the next remainder resets to 0 and then the pattern repeats again from 1. So we’re looking for a pattern where the remainders go up by 1, reset to 0 after the remainder = 3, and then repeat again from 1. This means the answer is B, 2, 3, 0, 1, 2 Luckily, Joanne's remaining eggs did not go unloved for long. Prime numbers The SAT loves to test students on prime numbers, so you should expect to see one question per test on prime numbers. Be sure to understand what they are and how to find them. A prime number is a number that is only divisible by two numbers- itself and 1. For example, is a prime number because $1 * $ is its only factor. ( is not evenly divisible by 2, 3, 4, 5, 6, 7, 8, 9, or 10). 12 is NOT a prime number, because its factors are 1, 2, 3, 4, 6, and 12. It has more factors than just itself and 1. 1 is NOT a prime number, because its only factor is 1. The only even prime number is 2. Questions about primes come up fairly often on the SAT and understanding that 2 (and only 2!) is a prime number will be invaluable for solving many of these. A prime number $x$ is squared and then added to a different prime number, $y$. Which of the following could be the final result? An even number An odd number A positive number A. I onlyB. II onlyC. III onlyD. I and III onlyE. I, II, and III Now this question relies on your knowledge of both number relationships and primes. You know that any number squared (the number times itself) will be an even number if the original number was even, and an odd number if the original number was odd. Why? Because an even * an even = an even, and an odd * an odd = an odd ($6 * 6 = 36$ $7 * 7 = 49$). Next, we are adding that square to another prime number. You’ll also remember that an even number + an odd number is odd, an odd number + an odd number is even, and an even number + an even number is even. Knowing that 2 is a prime number, let’s replace x with 2. $2^2 = 4$. Now if y is a different prime number (as stipulated in the question), it must be odd, because the only even prime number is 2. So let’s say $y = 3$. $4 + 3 = 7$. So the end result is odd. This means II is correct. But what if both x and y were odd prime numbers? So let’s say that $x = 3$ and $y = 5$. So $3^2 = 9$. $9 + 5 = 14$. So the end result is even. This means I is correct. Now, for option number III, our results show that it is possible to get a positive number result, since both our results were positive. This means the final answer is E, I, II, and III If you forgot that 2 was a prime number, you would have picked D, I and III only, because there would have been no possible way to get an odd number. Remembering that 2 is a prime number is the key to solving this question. Another typical prime number question on the SAT will ask you to identify how many prime numbers fall in a certain range of numbers. How many prime numbers are between 30 and 50, inclusive? A. TwoB. ThreeC. FourD. FiveE. Six This might seem intimidating or time-consuming, but I promise you do NOT need to memorize a list of prime numbers. First, eliminate all even numbers from the list, as you know the only even prime number is 2. Next, eliminate all numbers that end in 5. Any number that ends is 5 or 0 is divisible by 5. Now your list looks like this: 31, 33, 37, 39, 41, 43, 47, 49 This is much easier to work with, but we need to narrow it down further. (You could start using your calculator here, or you can do this by hand.) A way to see if a number is divisible by 3 is to add the digits together. If that number is 3 or divisible by 3, then the final result is divisible by 3. For example, the number 31 is NOT divisible by 3 because $3 + 1 = 4$, which is not divisible by 3. However 33 is divisible by 3 because $3 + 3 = 6$, which is divisible by 3. So we can now eliminate 33 ($3 + 3 = 6$) and 39 ($3 + 9 = 12$) from the list. We are left with 31, 37, 41, 43, 47, 49. Now, to make sure you try every necessary potential factor, take the square root of the number you are trying to determine is prime. Any integer equal to or less than the square root will be a potential factor, but you do not have to try any numbers higher. Why? Well let’s take 36 as an example. Its factors are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. But now look at the factor pairings. 1 36 2 18 3 12 4 9 6 6 (9 4) (12 3) (18 2) (36 1) After you get past 6, the numbers repeat. If you test out 4, you will know that 9 goes evenly into your larger number- no need to actually test 9 just to get 4 again! So all numbers less than or equal to a potential prime’s square root are the only potential factors you need to test. Going back to our list, we have 31, 37, 41, 43, 47, 49. Well the closest square root to 31 and 37 is 6. We already know that neither 2 nor 3 nor 5 factor evenly into 31 and 37. Neither do 4, or 6. You’re done. Both 31 and 37 must be prime. As for 41, 43, 47, and 49, the closest square root of these is 7. We already know that neither 2 nor 3 nor 5 factor evenly into 41, 43, 47, or 49. 7 is the exact square root of 49, so we know 49 is NOT a prime. As for 41, 43, and 47, neither 4 nor 6 nor 7 go into them evenly, so they are all prime. You are left with 31, 37, 41, 43, and 47. So your answer is D, there are five prime numbers (31, 37, 41, 43, and 47) between 30 and 50. Prime numbers, Prime Directive, either way I'm sure we'll live long and prosper. Absolute Values Absolute values are a concept that the SAT loves to use, as it is all too easy for students to make mistakes with absolute values. Expect to see one question on absolute values per test (though very rarely more than one). An absolute value is a representation of distance along a number line, forward or backwards. This means that an absolute value equation will always have two solutions. It also means that whatever is in the absolute value sign will be positive, as it represents distance along a number line and there is no such thing as a negative distance. An equation $|x + 3| = 14$, has two solutions: $x = $ $x = -17$ Why -17? Well $-17 + 3 = -14$ and, because it is an absolute value (and therefore a distance), the final answer becomes positive. So $|-14| = 14$ When you are presented with an absolute value, instead of doing the math in your head to find the negative and positive solution, rewrite the equation into two different equations. When presented with the above equation $|x + 3| = 14$, take away the absolute value sign and transform it into two equations- one with a positive solution and one with a negative solution. So $|x + 3| = 14$ becomes: $x + 3 = 14$ AND $x + 3 = -14$ Solve for $x$ $x = $ and $x = -17$ $|10 - k| = 3$ $|k - 5| = 8$. What is a value for $k$ that fulfills both equations above? We know that any given absolute value expression will have two solutions, so we must find the solution that each of these equations shares in common. For our first absolute value equation, we are trying to find the numbers for $k$ that, when subtracted from 10 will give us 3 and -3. That means our $k$ values will be 7 and 13. Why? Because $10 - 7 = 3$ and $10 - 13 = -3$ Now let's look at our second equation. We know that the two numbers for $k$ for $k - 5$ must give us both 8 and -8. This means our $k$ values will be 13 and -3. Why? Because $13 - 5 = 8$ and $-3 - 5 = -8$. 13 shows up as a solution for both problems, which means it is our answer. So our final answer is 13, this is the number for $k$ that can solve both equations. Consecutive Numbers Questions about consecutive numbers may or may not show up on your SAT. If they appear, it will be for a maximum of one question. Regardless, they are still an important concept for you to understand. Consecutive numbers are numbers that go continuously along the number line with a set distance between each number. So an example of positive, consecutive numbers would be: 4, 5, 6, 7, 8 An example of negative, consecutive numbers would be: -8, -7, -6, -5, -4 (Notice how the negative integers go from greatest to least- if you remember the basic guide to integers, this is because of how they lie on the number line in relation to 0) You can write unknown consecutive numbers out algebraically by assigning the first in the series a variable, $x$, and then continuing the sequence of adding 1 to each additional number. The sum of four positive, consecutive integers is 54. What is the first of these integers? If x is our first, unknown, integer in the sequence, so you can write all four numbers as: $x + (x + 1) + (x + 2) + (x + 3) = 54$ $4x + 6 = 54$ $4x = 48$ $x = 12$ So, because x is our first number in the sequence and $x= 12$, the first number in our sequence is 12. You may also be asked to find consecutive even or consecutive odd integers. This is the same as consecutive integers, only they are going up every other number instead of every number. This means there is a difference of two units between each number in the sequence instead of 1. An example of positive, consecutive even integers: 8, 10, 12, 14, 16 An example of positive, consecutive odd integers: 15, 17, 19, 21, 23 Both consecutive even or consecutive odd integers can be written out in sequence as: $x, x + 2, x + 4, x + 6$, etc. No matter if the beginning number is even or odd, the numbers in the sequence will always be two units apart. What is the median number in the sequence of five positive, consecutive odd integers whose sum is 185? $x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 185$ $5x + 20 = 185$ $5x = 165$ $x = 33$ So the first number in the sequence is 33. This means the full sequence is: 33, 35, 37, 39, 41 The median number in the sequence is 37. Bonus history lesson- Grover Cleveland is the only US president to have ever served two non-consecutive terms. Steps to Solving an SAT Integer Question Because SAT integer questions are so numerous and varied, there is no set way to approach them that is entirely separate from approaching other kinds of SAT math questions. But there are a few techniques that will help you approach your SAT integer questions (and by extension, most questions on SAT math). #1: Make sure the question requires an integer. If the question does NOT specify that you are looking for an integer, then any number- including decimals and fractions- are fair game. Always read the question carefully to make sure you are on the right track. #2: Use real numbers if you forget your integer rules. Forget whether positive, even consecutive integers should be written as $x + (x + 1)$ or $x + (x + 2)$? Test it out with real numbers! 14, 16, 18 are consecutive even integers. If $x = 14$, $16 = x + 2$, and $18 = x + 4$. This works for most all of your integer rules. Forget your exponent rules? Plug in real numbers! Forget whether an even * an even makes an even or an odd? Plug in real numbers! #3: Keep your work organized. Like with most SAT math questions, integer questions can seem more complex than they are, or will be presented to you in strange ways. Keep your work well organized and keep track of your values to make sure your answer is exactly what the question is asking for. Santa is magic and has to double-check his list. So make sure you double-check your work too! Test Your Knowledge 1. If $a^x * a^6 = a^24$ and $(a^3)^y = a^15$, what is the value of $x + y$? A. 9B. 12C. 23D. 30E. 36 2. If $48√48 = a√b$ where $a$ and $b$ are positive integers and $a b$, which of the following could be a value of $ab$? A. 48B. 96C. 192D. 576E. 768 3. What is the product of the smallest prime number that is greater than 50 and the greatest prime number that is less than 50? 4.If $j, k$, and $n$ are consecutive integers such that $0jkn$ and the units (ones) digit of the product $jn$ is 9, what is the units digit of $k$? A. 0B. 1C. 2D. 3E. 4 Answers: C, D, 2491, A Answer Explanations: 1. In this question, we are being asked both to multiply bases with exponents as well as take a base with an exponent to another exponent. Essentially, the question is testing us on whether or not we know our exponent rules. If we remember our exponent rules, then we know that we must add exponents when we are multiplying two of the same base together. So $a^x * a^6 = a^24$ = $a^{x + 6} = a^24$ $x + 6 = 24$ $x = 18$ We have our value for $x$. Now we must find our $y$. We also know that, when taking a base and exponent to another exponent, we must multiply the exponents. So $(a^3)^y = a^15$ = $a^{3 * y} = a^15$ $3 * y = 15$ $y = 5$ In the final step, we must add our $x$ and $y$ values together: $18 + 5 = 23$ So our final answer is C, 23. 2. We are starting with $48√48$ and we know we must reduce it. Why? Because we are told that our first $48 = a$ and our second $48 = b$ AND that $a b$. Right now our $a$ and $b$ are equal, but, by reducing the expression, we will be able to find an $a$ value that is greater than our $b$ So let's find all the factors of 48 to see if there are any perfect squares. 48 $1 * 48$ $2 * 24$ $3 * 16$ $4 * 12$ $6 * 8$ Two of these pairings have perfect squares. 16 is our largest perfect square, which means that it will be the number we must use to reduce $48√48$ down to its most reduced form. Though we are not explicitly asked to find the most reduced form of $48√48$, we can start there for now. So $48√48 = 48 * √16 * √3$ $48 * 4 *√3$ $192√3$ This means that our $a = 192$ and our $b = 3$, then: $ab = 192 * 3 = 576$ So our final answer is D, 576. (Special note: you'll notice how we are told to find one possible value for $ab$, not necessarily $ab$ when $48√48$ is at its most reduced. So if our above answer hadn't matched one of our answer options, we would have had to reduce $48√48$ only part way. $48√48 = 48 * √4 * √12$ $48 * 2 * √12$ $96√12$ This would make our $a = 96$ and our $b = 12$, meaning that our final answer for $ab$ would be $96 * 12 = 52$.) 3. This question requires us to be able to figure out which numbers are prime. Let us use the same methods we used during the above section on prime numbers. All prime numbers other than 2 will be odd and there is no prime number that ends in 5. So let's list the odd numbers (excluding ones that end in 5's) above and below 50. 41, 43, 47, 49, 51, 53, 57, 59 We are trying to find the ones closest to 50 on either side, so let's first test the highest number in the 40's. 49 is the perfect square of 7, which means it is divisible by more than just itself and 1. We can cross 49 off the list. 47 is not divisible by 3 because $7 + 4 = $ and is not divisible by 3. It is also not divisible by any even number (because an even * an even = an even), by 5, or by 7. This means it must be prime. (Why did we stop here? Remember that we only have to test potential factors up until the closest square root of the potential prime. $√47$ is between $6^2 = 36$ and $7^2 = 49$, so we tested 7 just to be safe. Once we saw that 7 could not go into 47, we proved that 47 is a prime.) 47 is our largest prime less than 50. Now let's test the smallest number greater than 50. 51 is odd, but $5 + 1 = 6$, which is divisible by 3. That means that 51 is also divisible by 3 and thus cannot be prime. 53 is not divisible by 3 because $5 + 3 = 8$, which is not divisible by 3. It is also not divisible by 5 or 7. Therefore it is prime. (Again, we stopped here because the closest square root to 53 is between 7 and 8. And 8 cannot be a prime factor because all of its multiples are even). This means our smallest prime less than 50 is 47 and our largest is 53. Now we just need to find the product of those two numbers. $47 * 53 = 2491$ Our final answer is 2491. 4. We are told that $j$, $k$, and $n$ are consecutive integers. We also know they are positive (because they are greater than 0) and that they go in ascending order, $j$ to $k$ to $n$. We are also told that $jn$ equals a number with a units digit of 9. So let's find all the ways to get a product of 9 with two numbers. $1 * 9$ $3 * 3$ The only way to get any number that ends in 9 (units digit 9) from the product of two numbers is in one of two ways: #1: Both the original numbers have a units digit of 3 #2: The two original numbers have units digits of 1 and 9, respectively. Now let's visualize positive consecutive integers. Positive consecutive integers must go up in order with a difference of 1 between each variable. So $j, k, n$ could look like any collection of three numbers along a consistent number line. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, , 12, 13, 14, 15, 16, etc. There is no possible way that the units digits of the first and last of three consecutive numbers could both be 3. Why? Because if one had a units digit of 3, the other would have to end in either 1 or 5. Take 13 as an example. If $j$ were 13, then $n$ would have to be 15. And if $n$ were 13, then $j$ would have to be . So we know that neither $j$ nor $n$ has a units digit of 3. Now let's see if there is a way that we can give $j$ and $n$ units digits of 1 and 9 (or 9 and 1). If $j$ were given a units digit of 1, $n$ would have a units digit of 3. Why? Picture $j$ as . $n$ would have to be 13, and $ * 13 = 143$, which means the units digit of their product is not 9. But what if $n$ was a number with a units digit of 1? $j$ would have a units digit of 9. Why? Picture $n$ as now. $j$ would be 9. $9 * = 99$. The units digit is 9. And if the last digit of $j$ is 9 and the numbers $j, k, \and n$ are consecutive, then $k$ has to end in 0. So our final answer is A, 0. The Take-Aways Integers and integer questions can be tricky for some students, as they often involve concepts not tested in high school level math classes (when’s the last time you dealt with integer remainders, for example?). But most integer questions are much simpler than they appear. If you know your definitions- integers, consecutive integers, absolute values, etc.- and you know how to pay attention to what the question is asking you to find, you’ll be able to solve most any integer question that comes your way. What’s Next? Whew! You’ve done your paces on integers, both basic and advanced. Now that you’ve tackled these foundational topics of the SAT math, make sure you’ve got a solid grasp of all the math topics covered by the SAT math section, so that you can take on the SAT with confidence. Find yourself running out of time on SAT math? Check out our article on how to buy yourself time and complete your SAT math problems before time’s up. Feeling overwhelmed? Start by figuring out your ideal score and check out how to improve a low SAT math score. Already have pretty good scores and looking to get a perfect 800 on SAT Math? Check out our article on how to get a perfect score written by a full SAT scorer. Want to improve your SAT score by 160 points? 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