GRE Integer Questions Practice Questions with Answers

Integers represent the foundation of the Quantitative Reasoning section, appearing in various formats from simple arithmetic to complex number theory problems. Success on the GRE requires a firm grasp of how whole numbers interact through addition, subtraction, multiplication, and division. By refining your skills with GRE integer questions, you can secure easy points on the test and build the logic necessary for more advanced algebraic concepts. Understanding the properties of odd and even numbers, prime factorization, and divisibility rules is essential for any high-scoring student.

Concept Explanation

An integer is any whole number—positive, negative, or zero—that does not contain a fractional or decimal component. In the context of the GRE, integers include the set $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. Key properties of integers that frequently appear on the exam include parity (odd vs. even), divisibility, and prime numbers. For example, an even number is any integer divisible by 2, while an odd number is not. It is vital to remember that 0 is an even integer. When multiplying integers, the product of two even numbers is even, two odd numbers is odd, and one of each is even. Additionally, the Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. This concept, known as prime factorization, is a powerful tool for solving GRE integer questions involving factors and multiples.

To effectively prepare for the quantitative section, you might find it helpful to use a GRE Prep hub to organize your study schedule. Many students also utilize an AI Question Generator to create custom drills that focus specifically on their weakest areas, such as remainder problems or consecutive integer sequences.

Solved Examples

1. Problem: If $x$ is an even integer and $y$ is an odd integer, which of the following must be an odd integer?
   (A) $xy$
   (B) $x + 2y$
   (C) $2x + y$
   (D) $3xy$
   Solution:
   1. Identify the properties: Even $\times$ Odd = Even. Therefore, $xy$ is even. (A and D are out).
   2. Analyze (B): $x$ (even) + $2y$ (even) = Even.
   3. Analyze (C): $2x$ (even) + $y$ (odd) = Odd.
   4. The correct answer is (C).
2. Problem: What is the greatest common factor (GCF) of 48 and 72?
   Solution:
   1. Find the prime factorization of 48: $48 = 2^4 \times 3$.
   2. Find the prime factorization of 72: $72 = 2^3 \times 3^2$.
   3. Identify the common prime factors with the lowest exponents: $2^3$ and $3^1$.
   4. Multiply them: $8 \times 3 = 24$. The GCF is 24.
3. Problem: If $n$ is an integer such that $10 < n < 20$, and $n$ is a prime number, what are the possible values for $n$?
   Solution:
   1. List integers between 10 and 20: 11, 12, 13, 14, 15, 16, 17, 18, 19.
   2. Eliminate even numbers: 11, 13, 15, 17, 19.
   3. Eliminate multiples of 3 or 5: 15 is divisible by 3 and 5.
   4. The remaining prime numbers are 11, 13, 17, and 19.

Practice Questions

1. If $k$ is an integer, and $5k$ is an odd integer, which of the following must be true about $k$?

2. What is the remainder when $7^{10}$ is divided by 10?

3. The sum of three consecutive integers is 72. What is the largest of these three integers?

4. How many prime numbers are there between 20 and 40?

5. If $x$ and $y$ are positive integers such that $x + y = 10$, what is the maximum possible value of $xy$?

6. Is the product of any four consecutive integers always divisible by 24?

7. If $m$ is an even integer and $n$ is an odd integer, is $(m+n)^2$ even or odd?

8. List all the factors of 60.

9. Quantity A: The number of distinct prime factors of 150. Quantity B: The number of distinct prime factors of 105.

10. If $n$ is a multiple of 6 and a multiple of 9, what is the smallest positive value of $n$?

Answers & Explanations

1. Answer: $k$ must be odd.
   Explanation: An odd number multiplied by another number results in an odd number only if both numbers are odd. Since 5 is odd, $k$ must also be odd.
2. Answer: 9.
   Explanation: Remainder when dividing by 10 is the units digit. Look for a pattern in powers of 7: $7^1=7, 7^2=49, 7^3=...3, 7^4=...1$. The pattern repeats every 4 powers. $10 \div 4 = 2$ remainder 2. The second number in the pattern is 9.
3. Answer: 25.
   Explanation: Let the integers be $x-1, x, x+1$. Their sum is $3x = 72$, so $x = 24$. The largest integer is $x+1 = 25$.
4. Answer: 4.
   Explanation: The prime numbers between 20 and 40 are 23, 29, 31, and 37.
5. Answer: 25.
   Explanation: For a fixed sum, the product is maximized when the numbers are as close as possible. $5 + 5 = 10$, and $5 \times 5 = 25$.
6. Answer: Yes.
   Explanation: In any four consecutive integers, at least one is a multiple of 4, another is a multiple of 3, and at least two are even. $4 \times 3 \times 2 = 24$.
7. Answer: Odd.
   Explanation: Even + Odd = Odd. An odd number squared (Odd $\times$ Odd) is always Odd.
8. Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
   Explanation: Pairs are (1,60), (2,30), (3,20), (4,15), (5,12), (6,10).
9. Answer: Quantity A is equal to Quantity B.
   Explanation: $150 = 2 \times 3 \times 5^2$ (3 distinct factors: 2, 3, 5). $105 = 3 \times 5 \times 7$ (3 distinct factors: 3, 5, 7). Both have 3.
10. Answer: 18.
    Explanation: This is the Least Common Multiple (LCM). $6 = 2 \times 3$, $9 = 3^2$. LCM = $2 \times 3^2 = 18$.

1. Which of the following is the only even prime number?

• A0
• B1
• C2
• D4

Frequently Asked Questions

Is zero considered a positive or negative integer?

Zero is neither positive nor negative, but it is an even integer and a whole number. It serves as the neutral element for addition in the set of all integers.

What is the difference between a factor and a multiple?

A factor is a number that divides into another number evenly without a remainder, whereas a multiple is the product of an integer and another integer. For example, 3 is a factor of 12, and 12 is a multiple of 3.

Are negative numbers allowed to be prime?

No, by standard mathematical definition, prime numbers must be integers greater than 1. Therefore, negative numbers, 0, and 1 are not prime.

How do you find the number of factors of a large integer quickly?

Find the prime factorization $p_1^a \times p_2^b \times p_3^c$, then add 1 to each exponent and multiply them: $(a+1)(b+1)(c+1)$. This formula accounts for all possible combinations of prime factors.

What are consecutive integers?

Consecutive integers are numbers that follow each other in order, each being 1 more than the previous one, such as 7, 8, and 9. They are often represented algebraically as $n, n+1, n+2$.