# Learn, Practice, and Ace GMAT Number Properties: Strategy Guide 5 (PDF)

# Manhattan GMAT Number Properties 5th Edition PDF Download - Introduction - What is the Manhattan GMAT Number Properties guide and why is it useful for GMAT test takers? - What are the main topics covered in the guide and how are they organized? - How to use the guide effectively for optimal results? - Divisibility and Primes - What are divisibility rules and how to apply them to different types of numbers? - What are prime numbers and how to identify them quickly? - What are prime factorization and greatest common factor (GCF)? - How to solve problems involving divisibility and primes on the GMAT? - Odds and Evens - What are odd and even numbers and how to recognize them? - What are the properties of odd and even numbers when performing arithmetic operations? - How to solve problems involving odds and evens on the GMAT? - Positives and Negatives - What are positive and negative numbers and how to compare them? - What are the properties of positive and negative numbers when performing arithmetic operations? - How to solve problems involving positives and negatives on the GMAT? - Consecutive Integers - What are consecutive integers and how to represent them algebraically? - What are the properties of consecutive integers when performing arithmetic operations? - How to solve problems involving consecutive integers on the GMAT? - Exponents - What are exponents and how to interpret them? - What are the properties of exponents when performing arithmetic operations? - How to solve problems involving exponents on the GMAT? - Roots - What are roots and how to interpret them? - What are the properties of roots when performing arithmetic operations? - How to solve problems involving roots on the GMAT? - Remainders - What are remainders and how to calculate them? - What are the properties of remainders when performing arithmetic operations? - How to solve problems involving remainders on the GMAT? - Absolute Value - What is absolute value and how to interpret it? - What are the properties of absolute value when performing arithmetic operations? - How to solve problems involving absolute value on the GMAT? - Number Lines - What is a number line and how to use it to visualize numbers and their relationships? - How to use number lines to solve problems involving inequalities, fractions, decimals, percents, ratios, rates, and proportions on the GMAT? - Practice Problems - A table with 15 practice problems from easy to hard level, covering all the topics discussed in the guide. - Each problem has a detailed explanation of the solution, highlighting the key concepts and strategies used. - Conclusion - A summary of the main points and takeaways from the guide. - A call to action for the readers to download the PDF version of the guide from a reliable source. - FAQs - Five frequently asked questions about the Manhattan GMAT Number Properties guide, such as: - Where can I find more practice problems on number properties for the GMAT? - How can I improve my speed and accuracy on number properties questions on the GMAT? - How can I avoid common traps and pitfalls on number properties questions on the GMAT? - How does number properties relate to other topics on the GMAT quant section? - How can I review my mistakes and learn from them on number properties questions on the GMAT? Now, based on this outline, here is the article I will write: # Manhattan GMAT Number Properties 5th Edition PDF Download If you are preparing for the Graduate Management Admission Test (GMAT), you know that mastering number properties is essential for scoring well on the quantitative section. Number properties questions test your ability to understand and manipulate various types of numbers, such as integers, fractions, decimals, percents, exponents, roots, etc. These questions can be tricky and time-consuming if you don't have a solid grasp of the concepts and strategies involved. That's why you need a reliable and comprehensive guide that can help you ace number properties questions on the GMAT. One such guide is the Manhattan GMAT Number Properties 5th Edition. This guide is part of the Manhattan GMAT Strategy Guide series, which covers all the topics tested on the GMAT quant section. The Number Properties guide provides a thorough analysis of the properties and rules of integers tested on the GMAT, covering both simple and complicated concepts. You will learn, practice, and master everything from prime products to perfect squares. In this article, we will give you an overview of the main topics covered in the Manhattan GMAT Number Properties guide, and how to use it effectively for optimal results. We will also provide you with a link to download the PDF version of the guide from a reliable source, so you can start studying right away. Let's get started! ## Divisibility and Primes The first topic covered in the Manhattan GMAT Number Properties guide is divisibility and primes. Divisibility rules are shortcuts that help you determine whether a number is divisible by another number without actually performing the division. For example, a number is divisible by 2 if its last digit is even, and a number is divisible by 3 if the sum of its digits is divisible by 3. Knowing these rules can save you time and avoid mistakes on the GMAT. Prime numbers are numbers that have exactly two factors: 1 and themselves. For example, 2, 3, 5, 7, 11, etc. are prime numbers, while 4, 6, 8, 9, etc. are not. Prime numbers play an important role in number properties questions on the GMAT, as they can help you break down complex numbers into simpler components. For example, you can use prime factorization to find the greatest common factor (GCF) of two or more numbers, which is the largest number that divides all of them evenly. The Manhattan GMAT Number Properties guide teaches you how to identify prime numbers quickly using various methods, such as divisibility tests, elimination techniques, and approximation strategies. You will also learn how to solve problems involving divisibility and primes on the GMAT using different approaches, such as plugging in numbers, picking numbers, working backwards, and testing cases. ## Odds and Evens The next topic covered in the Manhattan GMAT Number Properties guide is odds and evens. Odd and even numbers are integers that are either divisible by 2 or not. For example, -4, 0, 2, 6, etc. are even numbers, while -3, 1, 5, 7, etc. are odd numbers. Knowing how to recognize odd and even numbers can help you eliminate wrong answer choices on the GMAT. Odd and even numbers have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, adding two odd numbers always results in an even number, while multiplying two even numbers always results in an even number. Knowing these properties can help you simplify calculations and solve equations on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving odds and evens on the GMAT using different strategies, such as algebraic manipulation, logic and reasoning, pattern recognition, and estimation. ## Positives and Negatives The third topic covered in the Manhattan GMAT Number Properties guide is positives and negatives. Positive and negative numbers are integers that are either greater than zero or less than zero. For example, 1, 2, 3, etc. are positive numbers, while -1, -2, -3, etc. are negative numbers. Knowing how to compare positive and negative numbers can help you order them correctly on the GMAT. Positive and negative numbers have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, adding two positive numbers always results in a positive number, while multiplying two negative numbers always results in a positive number. Knowing these properties can help you avoid sign errors and solve inequalities on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving positives and negatives on the GMAT using different strategies, such as sign analysis, number line, and absolute value. ## Consecutive Integers The fourth topic covered in the Manhattan GMAT Number Properties guide is consecutive integers. Consecutive integers are integers that follow each other in order without any gaps. For example, -2, -1, 0, 1, 2, etc. are consecutive integers, while -3, -1, 1, 3, etc. are not. Knowing how to represent consecutive integers algebraically can help you express general rules and formulas on the GMAT. Consecutive integers have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, the sum of n consecutive integers is always divisible by n, while the product of n consecutive integers is always divisible by n factorial (n!). Knowing these properties can help you find patterns and shortcuts on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving consecutive integers on the GMAT using different strategies, such as grouping terms, factoring expressions, and applying formulas. ## Exponents The fifth topic covered in the Okay, here is the rest of the article: ## Exponents The fifth topic covered in the Manhattan GMAT Number Properties guide is exponents. Exponents are a way of expressing repeated multiplication of a number by itself. The number itself is called the base, and the exponent indicates how many times to multiply the base by itself. For example, 2^3 means 2 multiplied by itself 3 times, or 2 x 2 x 2 = 8. Exponents have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, when multiplying two numbers with the same base, you can add their exponents. When dividing two numbers with the same base, you can subtract their exponents. When raising a number with an exponent to another exponent, you can multiply their exponents. Knowing these properties can help you simplify expressions and solve equations on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving exponents on the GMAT using different strategies, such as factorization, distribution, substitution, and estimation. ## Roots The sixth topic covered in the Manhattan GMAT Number Properties guide is roots. Roots are the inverse operation of exponents. They tell you what number, when raised to a certain power, equals another number. For example, the square root of 9 is 3, because 3^2 = 9. The cube root of 27 is 3, because 3^3 = 27. Roots have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, when multiplying or dividing two numbers with the same root, you can combine them under one root sign. When raising a number with a root to an exponent, you can multiply the root and the exponent. Knowing these properties can help you simplify expressions and solve equations on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving roots on the GMAT using different strategies, such as rationalization, approximation, and elimination. ## Remainders The seventh topic covered in the Manhattan GMAT Number Properties guide is remainders. Remainders are what is left over when you divide one number by another number that does not divide it evenly. For example, when you divide 7 by 3, you get a quotient of 2 and a remainder of 1. You can write this as 7 = 3 x 2 + 1. Remainders have certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, when adding or subtracting two numbers with remainders, you can add or subtract their remainders separately. When multiplying or dividing two numbers with remainders, you can multiply or divide their remainders separately and adjust for any extra factors. Knowing these properties can help you find patterns and relationships on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving remainders on the GMAT using different strategies, such as modular arithmetic, divisibility rules, and prime factorization. ## Absolute Value The eighth topic covered in the Manhattan GMAT Number Properties guide is absolute value. Absolute value is a measure of how far a number is from zero on the number line. It is always positive or zero, and it is denoted by two vertical bars around the number. For example, the absolute value of -5 is 5, and the absolute value of 0 is 0. Absolute value has certain properties when performing arithmetic operations, such as addition, subtraction, multiplication, and division. For example, when adding or subtracting two numbers with absolute values, you can use the triangle inequality to find bounds for their sum or difference. When multiplying or dividing two numbers with absolute values, you can multiply or divide their absolute values separately. Knowing these properties can help you solve inequalities and equations on the GMAT. The Manhattan GMAT Number Properties guide teaches you how to apply these properties to solve problems involving absolute value on the GMAT using different strategies, such as squaring both sides, splitting cases, and testing values. ## Number Lines The ninth and final topic covered in the Manhattan GMAT Number Properties guide is number lines. Number lines are visual representations of numbers and their relationships on a horizontal line. They can help you compare, order, and manipulate numbers on the GMAT. Number lines can be used to solve problems involving inequalities, fractions, decimals, percents, ratios, rates, and proportions on the GMAT. For example, you can use number lines to find the midpoint or average of two numbers, to find the distance or difference between two numbers, to find the fraction or percent of a number, to find the ratio or proportion of two numbers, or to find the rate or speed of a number. The Manhattan GMAT Number Properties guide teaches you how to use number lines to solve problems on the GMAT using different strategies, such as plotting points, drawing segments, shifting positions, and scaling lengths. ## Practice Problems To help you practice and master the concepts and strategies discussed in the guide, the Manhattan GMAT Number Properties guide provides you with 15 practice problems from easy to hard level, covering all the topics in the guide. Each problem has a detailed explanation of the solution, highlighting the key concepts and strategies used. Here is a table with the practice problems and their solutions: Problem Solution ------------------- If x is an integer, which of the following must be an even integer? (A) x + 1 (B) x + 2 (C) x^2 + 1 (D) x^2 + 2 (E) x^2 + x (E) x^2 + x. To find out which expression must be even, we can test different values of x. If x is even, then x + 1 and x^2 + 1 are odd, while x + 2, x^2 + 2, and x^2 + x are even. If x is odd, then x + 1 and x^2 + 1 are even, while x + 2 and x^2 + 2 are odd. However, x^2 + x is still even, because it is the product of two odd numbers. Therefore, no matter what value of x we choose, x^2 + x must be an even integer. If n is a positive integer and n^3 is divisible by 24, what is the smallest possible value of n? (A) 2 (B) 3 (C) 4 (D) 6 (E) 8 (C) 4. To find the smallest possible value of n, we need to find the smallest possible value of n^3 that is divisible by 24. We can use prime factorization to break down both numbers into their prime factors. n^3 = n * n * n and 24 = 2 * 2 * 2 * 3. For n^3 to be divisible by 24, it must have at least three factors of 2 and one factor of 3. The smallest number that satisfies this condition is n = 4, because 4 = 2 * 2 and 4^3 = 64 = 2 * 2 * 2 * 2 * 2 * 2. Any smaller value of n would not have enough factors of 2 or 3 in its cube. If a and b are positive integers such that a/b = 98.765, what is the smallest possible value of b? (A) 11 (B) 37 (C) 41 (D) 101 (E) It cannot be determined from the information given. (D) 101. To find the smallest possible value of b, we need to find the simplest form of the fraction a/b. We can do this by converting the decimal part of a/b into a fraction and reducing it to its lowest terms. The decimal part of a/b is .765, which can be written as Okay, here is the rest of the article: Problem Solution ------------------- If x is an integer, which of the following must be an even integer? (A) x + 1 (B) x + 2 (C) x^2 + 1 (D) x^2 + 2 (E) x^2 + x (E) x^2 + x. To find out which expression must be even, we can test different values of x. If x is even, then x + 1 and x^2 + 1 are odd, while x + 2, x^2 + 2, and x^2 + x are even. If x is odd, then x + 1 and x^2 + 1 are even, while x + 2 and x^2 + 2 are odd. However, x^2 + x is still even, because it is the product of two odd numbers. Therefore, no matter what value of x we choose, x^2 + x must be an even integer. If n is a positive integer and n^3 is divisible by 24, what is the smallest possible value of n? (A) 2 (B) 3 (C) 4 (D) 6 (E) 8 (C) 4. To find the smallest possible value of n, we need to find the smallest possible value of n^3 that is divisible by 24. We can use prime factorization to break down both numbers into their prime factors. n^3 = n * n * n and 24 = 2 * 2 * 2 * 3. For n^3 to be divisible by 24, it must have at least three factors of 2 and one factor of 3. The smallest number that satisfies this condition is n = 4, because 4 = 2 * 2 and 4^3 = 64 = 2 * 2 * 2 * 2 * 2 * 2. Any smaller value of n would not have enough factors of 2 or 3 in its cube. If a and b are positive integers such that a/b = 98.765, what is the smallest possible value of b? (A) 11 (B) 37 (C) 41 (D) 101 (E) It cannot be determined from the information given. (D) 101. To find the smallest possible value of b, we need to find the simplest form of the fraction a/b. We can do this by converting the decimal part of a/b into a fraction and reducing it to its lowest terms. The decimal part of a/b is .765, which can be written as 765/1000. To reduce this fraction, we need to divide both the numerator and the denominator by their greatest common factor (GCF). The GCF of 765 and 1000 is 5, so we divide both by 5 to get 153/200. We can repeat this process until we cannot reduce the fraction any further. The GCF of 153 and 200 is 1, so we cannot reduce any more. Therefore, 153/200 is the simplest form of 765/1000. Now we can write a/b as a mixed number: a/b = 98 + (153/200). To convert this mixed number into an improper fraction, we need to multiply the whole number part by the denominator and add it to the numerator: a/b = (98 * 200 + 153)/200 = 19753/200. To find the smallest possible value of b, we need to find the smallest possible value of the denominator in this fraction. We can do this by finding the prime factorization of both the numerator and the denominator: 19753/200 = (3 * 7 * 941)/(2 * 2 * 5 *5). To reduce this fraction to its lowest terms, we need to cancel out any common factors in both the numerator and the denominator. However, there are no common factors in this case, so this fraction is already in its lowest terms. Therefore, 200 is the smallest possible value of b that makes a/b equal to 98.765. ## Conclusion We have covered the main topics and strategies that you need to know for GMAT Number Properties questions in this article. You have learned how to deal with divisibility and primes, odds and evens, positives and negatives, consecu