GRE Number Properties Questions Practice Questions with Answers

Approximately 25% of the Quantitative Reasoning section on the GRE focuses on the fundamental characteristics of integers and real numbers. Because the test makers prioritize logic over complex calculation, understanding GRE Number Properties Questions is essential for achieving a high score. These questions evaluate your grasp of divisibility, prime numbers, even and odd integers, and the behavior of remainders. By utilizing resources like GRE Prep, students can build a solid foundation in these mathematical rules to solve problems more efficiently during the exam.

Concept Explanation

Number properties refer to the specific rules and characteristics that govern integers, fractions, and real numbers, such as divisibility, parity, and primality. To excel at GRE Number Properties Questions, you must understand how these rules interact. For instance, the product of an even and an odd integer is always even, and any integer ending in 0 or 5 is divisible by 5. Prime numbers are integers greater than 1 that have only two factors: 1 and themselves. Common topics include the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 is either prime or a unique product of primes. Additionally, understanding remainders using the formula $\text{Dividend} = ( \text{Divisor} \times \text{Quotient}) + \text{Remainder}$ is vital for solving modular arithmetic problems. Many students find that using an AI Question Generator helps them see the various ways these concepts are tested in Quantitative Comparison and Problem Solving formats.

Solved Examples

Review these examples to see how number property rules are applied to GRE-style problems.

1. Example: If $n$ is an odd integer, which of the following must be even: $n^2$, $n+2$, or $3n+1$?
   1. Identify the rule for odd integers: $\text{odd} \times \text{odd} = \text{odd}$. Therefore, $n^2$ is odd.
   2. Identify the rule for addition: $\text{odd} + \text{even} = \text{odd}$. Since 2 is even, $n+2$ is odd.
   3. Evaluate $3n+1$: $\text{odd} \times \text{odd} = \text{odd}$, so $3n$ is odd. Then, $\text{odd} + \text{odd} = \text{even}$. So, $3n+1$ is even.
   4. The answer is $3n+1$.
2. Example: What is the greatest common factor (GCF) of 48 and 72?
   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 lowest power of each common prime factor: $2^3$ and $3^1$.
   4. Multiply these: $8 \times 3 = 24$. The GCF is 24.
3. Example: If $x$ is a positive integer and the remainder when $2x$ is divided by 4 is 0, what can be concluded about $x$?
   1. The condition implies $2x$ is a multiple of 4.
   2. Set up the equation: $2x = 4k$, where $k$ is an integer.
   3. Divide both sides by 2: $x = 2k$.
   4. This means $x$ must be an even integer.

Practice Questions

1. If $x$ and $y$ are even integers and $z$ is an odd integer, which of the following must be odd?
   (A) $xy + z$
   (B) $x + y + z + 1$
   (C) $xyz$
   (D) $x + z$
2. How many prime numbers are there between 20 and 40?
3. If $n$ is an integer and $n^2$ is divisible by 36, what is the smallest possible positive value of $n$?

4. Quantity A: The number of distinct prime factors of 60.
   Quantity B: The number of distinct prime factors of 90.
5. If $k$ is a positive integer such that $\frac{k}{5}$ has a remainder of 3, what is the remainder when $2k$ is divided by 5?
6. Is the product of any five consecutive integers always divisible by 120?
7. If $p$ is a prime number greater than 3, what is the remainder when $p^2$ is divided by 3?
8. If $x$ is a multiple of 6 and $y$ is a multiple of 9, then $x + y$ must be a multiple of which of the following?
   (A) 15
   (B) 18
   (C) 3
   (D) 54
9. What is the units digit of $7^{25}$?
10. If $a$ and $b$ are positive integers and $a$ is a factor of $b$, which of the following must be an integer?
    (A) $\frac{a}{b}$
    (B) $\frac{b}{a}$
    (C) $\frac{a+b}{b}$
    (D) $\sqrt{ab}$

Answers & Explanations

1. (A) and (D): Both are correct in this context. $xy$ is $\text{even} \times \text{even} = \text{even}$. $\text{Even} + \text{odd} (z) = \text{odd}$. Also, $x + z$ is $\text{even} + \text{odd} = \text{odd}$. In a multiple-choice GRE question, only one would be provided.
2. 4: The prime numbers between 20 and 40 are 23, 29, 31, and 37.
3. 6: If $n^2$ is divisible by 36, then $n^2 = 36, 72, 108, 144, \dots$. For $n$ to be an integer, $n^2$ must be a perfect square. The smallest perfect square divisible by 36 is 36 itself. Thus, $n = \sqrt{36} = 6$.
4. Equal: Prime factorization of $60 = 2^2 \times 3 \times 5$ (3 distinct factors: 2, 3, 5). Prime factorization of $90 = 2 \times 3^2 \times 5$ (3 distinct factors: 2, 3, 5).
5. 1: If $k = 5q + 3$, then $2k = 2(5q + 3) = 10q + 6$. Dividing $10q + 6$ by 5 gives a remainder of 1 (since 6 divided by 5 leaves 1).
6. Yes: The product of $n$ consecutive integers is always divisible by $n!$. Since $5! = 120$, the product of 5 consecutive integers is always divisible by 120.
7. 1: Any prime $p > 3$ can be written as $3k+1$ or $3k+2$. Squaring these: $(3k+1)^2 = 9k^2 + 6k + 1$ (rem 1) or $(3k+2)^2 = 9k^2 + 12k + 4 = 9k^2 + 12k + 3 + 1$ (rem 1).
8. (C) 3: $x = 6m = 3(2m)$ and $y = 9n = 3(3n)$. Their sum $3(2m + 3n)$ is always a multiple of 3.
9. 7: The units digits of powers of 7 follow a pattern: $7^1=7, 7^2=9, 7^3=3, 7^4=1$. The cycle length is 4. $25 \div 4$ leaves a remainder of 1, so the units digit is the same as $7^1$, which is 7.
10. (B) $\frac{b}{a}$: By definition, if $a$ is a factor of $b$, then $b$ is divisible by $a$, making $\frac{b}{a}$ an integer.

1. If \( x \) is an even integer and \( y \) is an odd integer, which of the following must be an odd integer?

• A\( x + y \)
• B\( x \times y \)
• C\( 2x + y + 1 \)
• D\( x^y \)

Frequently Asked Questions

Is 1 a prime number on the GRE?

No, 1 is not considered a prime number on the GRE or in standard mathematics because a prime must have exactly two distinct factors. Prime numbers start at 2, which is also the only even prime number.

What is the difference between a factor and a multiple?

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

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

To find the total number of factors, determine the prime factorization $a^p \times b^q \times c^r$, then add 1 to each exponent and multiply them: $(p+1)(q+1)(r+1)$. This method is much faster than listing factors manually.

Are negative numbers considered even or odd?

Yes, parity (being even or odd) applies to all integers, including negative ones and zero. Zero is even, -1 is odd, and -2 is even, following the standard alternating pattern on the number line.

What does "divisible" mean in GRE questions?

In the context of the GRE, "divisible" means that when one integer is divided by another, the result is an integer and the remainder is zero. This concept is central to many quantitative reasoning logic puzzles.