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I bought Nova's GRE Math Bible, I think the book is all you need for getting on quant. BUT you have to make sure you understand everything in the book, it is . The Nova Math Bible, if I am not mistaken is based on the old format, so you will not come across the numeric entry/multiple answer questions, which are on the. NOVA GRE Preparation Course Math: Twenty-two chapters provide comprehensive review of all the math properties you need for the GRE. Format: pdf.
Since 2 is an integer, the answer is B. Choice A: Choice B: Choice C: Choice D: Choice E: The answer is B. Choose any five consecutive integers, say, —2, —1, 0, 1 and 2. We chose these particular numbers to make the calculation as easy as possible. But any five consecutive integers will do. For example, 1, 2, 3, 4, and 5. Now, add 3 to the set to form 6 consecutive integers: Substitution 19 Method II without substitution: Hard 4.
Here, the remainder r equals 3. Here, the remainder is 6. Possible answer. Hence, the answer is B. Since we are dividing by 2n not by n , the remainder when divided by 2n is 2r. This is true for all 1 p 2. This is true for any 2 p 3.
The answer is E. We are given that k is a positive integer and m is a positive integer less than We are also given that In short, p must be a positive integer divisible by 9. Substitution 21 Substitution Quantitative Comparisons: You must also check negative numbers, fractions, 0, and 1 because they often give results different from those of positive whole numbers. Determine which of the two expressions below is larger, whether they are equal, or whether there is not enough information to decide.
In this case, Column B is larger. In this case, the two columns are equal. Hence, the answer is D —not enough information to decide. If, as above, you get a certain answer when a particular number is substituted and a different answer when another number is substituted Double Case , then the answer is D —not enough information to decide.
Let x denote the greatest integer less than or equal to x. For example: Now, which column below is larger? In this case, Column A equals Column B. In this case, the two columns are again equal. Thus, in this case Column B is larger. This is a double case.
Problem Set B: Column A m 0 Column B m10 m Medium 3. Column A Column B ab2 a2b For all numbers x, x denotes the value of x 3 rounded to the nearest multiple of ten. Column A 0 x 2 Column B x2 x In this case, Column A is larger. This is a double case and therefore the answer is D.
This is a double case, and the answer is D. Medium 3. This covers the three types of negative numbers, so we can confidently conclude the answer is A. There is only one type of number between —1 and 0—negative fractions. Hence, Column A is larger, and the answer is A.
This is a double case and the answer is D. In this case, the columns are unequal. If x and y are positive, then Column B is positive and therefore larger than zero. If x and y are negative, then Column B is still positive since a negative divided by a negative yields a positive.
This covers all possible signs for x and y. In this case, the columns are equal. In this case, the columns are not equal and therefore the answer is D.
Sometimes instead of making up numbers to substitute into the problem, we can use the actual answer-choices. Method II This problem can also be solved by factoring. Substitution 25 Problem Set C: Use the method of Plugging In to solve the following problems. Which one of the following is the solution of the system of equations given? The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. This happens only when x is even.
Hence, x must be even. Since 16 is the only even answer-choice, the answer is E. Substitution 27 4. Since this value of x satisfies the equation, the answer is A.
We have the second solution in choice A , so the answer is A. Hard 5. So, reject the choice. So, select the choice. Defined Functions Defined functions are very common on the GRE, and at first most students struggle with them.
Yet, once you get used to them, defined functions can be some of the easiest problems on the test. In this type of problem, you will be given a symbol and a property that defines the symbol.
Some examples will illustrate. The integers between —8 and 2 are: For any positive integer n, n! What is the value of 3! As defined, 3! Hence, 3! If the value of 1 is 1, then 3 equals which one of the following? Defined Functions 29 You may be wondering how defined functions differ from the functions, f x , you studied in Intermediate Algebra and more advanced math courses.
They are the same old concept you dealt with in your math classes. Once you realize that defined functions are evaluated and manipulated just as regular functions, they become much less daunting. Problem Set D: Medium 1. Column A For any positive integer n, n! Column B 1!
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First, subtracting the numbers in the parentheses yields Column A Column B 1! As defined, 2! Hence, in Column B, we have 2! Hence, 1! Also, 9! Hence, in Column A, we have 1! Hence, Column A equals 7. So, 2 is the only even factor. Hence, the answer is A. The only factors of a prime number are 1 and itself. Now, the factors of ab are 1, a, b, and ab itself.
Hence, the total number of factors of ab is 4. Hence, Column B also equals 4. Since the columns are equal, the answer is C.
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There is exactly one multiple of 3 in every three consecutive positive integers. Also, at least one of the three numbers must be an even number. Method II: We can rewrite this problem using ordinary function notation. Multiplying both columns by mn to clear fractions yields n2 — m2 m2 — n2 Adding n2 and m2 to both columns yields 2n2 2m2 Finally, dividing both columns by 2 yields n2 m2 Since both m and n are positive and m n, m2 n2.
The numbers and are themselves multiples of 3. Also, a multiple of 3 exists once in every three consecutive integers.
Hence, Column A is greater than Column B by 1 unit. But for now, we need a brief review of these concepts for many of the problems that follow. To compare two fractions, cross-multiply.
The larger product will be on the same side as the 33 larger fraction. Given 5 6 vs. Now 36 is larger than 35, so 6 7 is larger than 5 6. Taking the square root of a fraction between 0 and 1 makes it larger. This is not true for fractions greater than 1. Squaring a fraction between 0 and 1 makes it smaller. This mistake is often seen in the following form: Memorize the following factoring formulas—they occur frequently on the GRE.
Know these rules for radicals: Pythagorean Theorem For right triangles only: What is the area of the triangle to the right? The answer is A. When parallel lines are cut by a transversal, three important angle relationships are formed: Alternate interior angles are equal. In a triangle, an exterior angle is equal to the sum of its remote interior angles and therefore greater than either of them.
A central angle has by definition the same measure as its intercepted arc. Math Notes 35 An inscribed angle has one-half the measure of its intercepted arc. In the triangle to the right, what is the degree measure of angle c? Hence, the answer is C. To find the percentage increase, find the absolute increase and divide by the original amount.
Systems of simultaneous equations can most often be solved by merely adding or subtracting the equations. Merely subtract the second equation from the first: When counting elements that are in overlapping sets, the total number will equal the number in one group plus the number in the other group minus the number common to both groups.
Venn diagrams are very helpful with these problems. If in a certain school 20 students are taking math and 10 are taking history and 7 are taking both, how many students are taking either math or history? History Math 10 7 20 Both History and Math By the principle stated above, we add 10 and 20 and then subtract 7 from the result. The number of integers between two integers inclusive is one more than their difference.
To see this more clearly, choose smaller numbers, say, 9 and The difference between 9 and 11 is 2. But there are three numbers between them inclusive—9, 10, and 11—one more than their difference. Rounding Off: But if the following digit is greater than or equal to five, then the preceding digit is increased by one. This notation has the following form: Students often use 6 as the exponent in the above example because there are 6 zeros.
At first, students often struggle with these problems since they have forgotten many of the basic properties of arithmetic. The remainder is 57 when a number is divided by 10, What is the remainder when the 37 same number is divided by 1,? A number n is even if the remainder is zero when n is divided by 2: A number n is odd if the remainder is one when n is divided by 2: This is possible only when n is even.
The integer zero is neither positive nor negative, but it is even: A prime number is an integer that is divisible only by itself and 1. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,. A number is divisible by 3 if the sum of its digits is divisible by 3. A common multiple is a multiple of two or more integers.
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For example, some common multiples of 2 and 5 are 0, 10, 20, 40, and The least common multiple LCM of two integers is the smallest positive integer that is a multiple of both. For example, the LCM of 4 and 10 is The standard method of calculating the LCM is to prime factor the numbers and then form a product by selecting each factor the greatest number of times it occurs.
Number Theory 39 A shortcut for finding the LCM is to just keep adding the largest number to itself until the other numbers divide into it evenly. For 4 and 10, we would add 10 to itself: Since 4 divides evenly in to 20, the LCM is For 8, 36, and 54, we would add 54 to itself: Since both 8 and 36 divide evenly into , the LCM is The absolute value of a number, , is always positive.
In other words, the absolute value symbol eliminates negative signs. Caution, the absolute value symbol acts only on what is inside the symbol,. Here, only the negative sign inside the absolute value symbol but outside the parentheses is eliminated.
The number of prime numbers divisible by 2 plus the number of prime numbers divisible by 3 is A 0 B 1 C 2 D 3 E 4 A prime number is divisible by no other numbers, but itself and 1. Hence, the only prime number divisible by 2 is 2 itself; and the only prime number divisible by 3 is 3 itself. Hence, The number of prime numbers divisible by 2 is one, and the number of prime numbers divisible by 3 is one.
Example 4: The product quotient of positive numbers is positive. The product quotient of a positive number and a negative number is negative. The product quotient of an even number of negative numbers is positive.
The sum of negative numbers is negative. But recall that subtraction is defined as negative addition. A number raised to an even exponent is greater than or equal to zero. If x is divisible by both 3 and 4, then the number x must be a multiple of which one of the following? The last digit of the positive even number n equals the last digit of n2. Which one of the following could be n?
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Which one of the following is divisible by both 2 and 3? A B C D E 4. Column A Column B The number of prime numbers divisible by 2 The number of prime numbers divisible by 3.
Which one of the following equals the product of exactly two prime numbers? Which one of the following equals the product of the smallest prime number greater than 21 and the largest prime number less than 16? Column B The two-digit number ab 40 8. Number Theory 41 Medium 9. Column A X is a 3-digit number and Y is a 4- digit number. All the digits of X are greater than 4, and all the digits of Y are less than 5. Column A Column B The least number divisible by 2, 3, 4, 5, and 6 The least number that is a multiple of 2, 3, 4, 5 and 6 The number 3 divides a with a result of b and a remainder of 2.
The number 3 divides b with a result of 2 and a remainder of 1. What is the value of a?
The remainder when the positive integer m is divided by 7 is x. Which one of the following could m equal? Which one of the following choices does not equal any of the other choices? Each of the two positive integers a and b ends with the digit 2.
With which one of the following numbers does a — b end? Column A x is a two-digit number. The digits of the number differ by 6, and the squares of the digits differ by Column B x 60 If each of the three nonzero numbers a, b, and c is divisible by 3, then abc must be divisible by which one of the following the numbers?
How many positive five-digit numbers can be formed with the digits 0, 3, and 5? If n is a positive integer, which one of the following numbers must have a remainder of 3 when divided by any of the numbers 4, 5, and 6? Column A Column B Least common multiple of the two positive integers m and n mn The number is divisible by both 6 and 8.
Which one of the following is the first integer larger than that is also divisible by both 6 and 8? A B C D E If the least common multiple of m and n is 24, then what is the first integer larger than that is divisible by both m and n? Column A Column B The first number larger than that is a multiple of both 6 and 8 Column A a, b, and c are consecutive integers in increasing order of size.
Number Theory 43 How many 3-digit numbers do not have an even digit or a zero? Column A The digits of a two-digit number x differ by 4. Column B The positive difference between the squares of the digits of x 15 Hard Which one of the following is the minimum value of the sum of two integers whose product is 36?
Column A m and n are two positive integers. Column B m n Column B The positive two-digit number ab The positive two-digit number ba Which one of the following numbers was deleted? A set has exactly five consecutive positive integers starting with 1. What is the percentage decrease in the average of the numbers when the greatest one of the numbers is removed from the set?
What is the maximum value of m such that 7m divides into 14! Column A A palindrome number is a number that reads the same forward or backward. For example, is a palindrome number. Column B Smallest palindrome number greater than Smallest palindrome greater than Very Hard The positive integers m and n leave remainders of 2 and 3, respectively, when divided by 6.
What is the remainder when m — n is divided by 6? If m n, then what is the remainder when mn divided by 6? N umber Theory 45 Which one of the following expressions is not odd? How many positive integers less than can be formed using the numbers 1, 2, 3, and 5 for the digits?
What is the remainder when 37 is divided by 8? Column A The positive integers m and n leave a remainder of 2 and 3, respectively, when divided by 6. We are given that x is divisible by 3 and 4.
Hence, x must be a common multiple of 3 and 4. The least common multiple of 3 and 4 is So, x is a multiple of Numbers ending with 0, 1, 5, or 6 will have their squares also ending with the same digit. Among the four numbers 0, 1, 5, or 6, even numbers only end with 0 or 6.
Choice D has one such number. A number divisible by 2 ends with one of the digits 0, 2, 4, 6, or 8. If a number is divisible by 3, then the sum of its digits is also divisible by 3. Hence, a number divisible by both 2 and 3 will follow both of the above rules. Choices A and C do not end with an even digit. Hence, eliminate them.
Also, the last digit is 6.
Hence, choice B is correct. Hence, reject the two choices. A prime number is divisible by no other numbers, but itself and 1.
Since the number of primes in each column is 1, the answer is C. The product of more than two primes. The smallest prime number greater than 21 is 23, and the largest prime number less than 16 is Suppose b equals 0. Here, Column A is less than Column B, which equals Now, suppose b equals 1.
Here Column A is greater than Column B, which equals Hence, we have a double case, and the answer is D. Number Theory 47 8. The last digit of the number in Column A is 2, and the last digit of the number in Column B is also 2. Hence, both numbers raised to the same power here 56 should end with the same digit. So, should end with the same digit as A small example: Medium 9. Now, p is a positive integer only when 0. Now, 0. This happens only when q is a multiple of Any of these values is greater than Column A: Since all the digits of the 3-digit number X are greater than 4, each digit must be greater than or equal to 5.
Column B: Since all the digits in the 4-digit number Y are less than 5, each digit must be less than or equal to 4.
Also, the minimum value of the sum of the digits of a 4-digit number is 1 for example, for , the sum of the digits is 1. Hence, we cannot know which column is greater. Any number that is divisible by the five numbers 2, 3, 4, 5, and 6 must also be a multiple of all five numbers.
So, Column A refers to the set of numbers that are divisible by 2, 3, 4, 5, and 6 and Column B refers to the set of numbers that are a multiple of 2, 3, 4, 5 and 6 refer to the same set.
Hence, the least values of the two columns are the same. Accept the choice. Reject choices A and B. Reject choice C. Reject choice D.
Since each of the two integers a and b ends with the same digit, the difference of the two numbers ends with 0. Suppose a and b are the two digits of the number x, and let a represent the greater of the two. Hence, the two digits are 2 and 8. They can be arranged in any order. Hence, 28 and 82 are two feasible solutions. Now, 28 is less than Column B, and 82 is greater than Column B. Since each one of the three numbers a, b, and c is divisible by 3, the numbers can be represented as 3p, 3q, and 3r, respectively, where p, q, and r are integers.
Since p, q, and r are integers, pqr is an integer and therefore abc is divisible by Let the digits of the five-digit positive number be represented by 5 compartments: Each of the last four compartments can be filled in 3 ways with any one of the numbers 0, 3 and 5. The first compartment can be filled in only 2 ways with only 3 and 5, not with 0, because placing 0 in the first compartment would yield a number with fewer than 5 digits.
Let m be a number that has a remainder of 3 when divided by any of the numbers 4, 5, and 6. Then m — 3 must be exactly divisible by all three numbers. Hence, m — 3 must be a multiple of the Least Common Multiple of the numbers 4, 5, and 6. Hence, the answer is E. We can also subtract 3 from each answer-choice, and the correct answer will be divisible by Hence, correct.
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Number Theory 49 Suppose the two positive integers m and n do not have a common factor, apart from 1. Then the LCM of m and n is mn. Now, suppose the two integers m and n have at least one common factor other than 1. Then the LCM uses the common factors only once, unlike mn. Hence, the LCM is less than mn. Although the GRE is a difficult test, it is a very learnable test. Twenty-two chapters provide comprehensive review of all the math properties you need for the GRE.
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