GRE Arithmetic Practice Test Practice Questions with Answers

Arithmetic accounts for approximately one-third of the Quantitative Reasoning section on the GRE, testing your fundamental understanding of number properties and operations. Success on the exam requires more than just basic calculation; it demands the ability to apply logic to integers, fractions, percent changes, and sequences. This GRE Arithmetic Practice Test is designed to refine these skills through targeted exercises and conceptual review.

By engaging with a variety of problem types, students can identify patterns that frequently appear in the official Educational Testing Service (ETS) assessments. Utilizing a structured GRE Prep strategy ensures that you are not just memorizing formulas, but developing the quantitative intuition necessary for a high score. Whether you are dealing with prime factorization or complex ratios, consistent practice remains the most effective way to reduce anxiety and increase speed during the actual test.

Concept Explanation

GRE Arithmetic is the study of real numbers and the basic operations—addition, subtraction, multiplication, and division—along with their properties and applications. The core syllabus includes integers (even/odd, prime numbers, divisibility), fractions, decimals, percentages, ratios, and number lines. Unlike higher-level algebra or geometry, arithmetic focuses on the inherent characteristics of numbers. For instance, understanding that the product of two odd integers is always odd, or that a percentage increase followed by a percentage decrease of the same value results in a net loss, are critical logical shortcuts. The exam often presents these concepts through Quantitative Comparison questions, where you must determine the relationship between two values, or Multiple-Choice questions that require precise computation. Mastery involves not just finding an answer, but understanding the underlying number theory to solve problems efficiently.

Solved Examples

1. Divisibility and Remainders: If $n$ is an integer and $n^2$ is divisible by 72, what is the smallest possible value of $n$?

1. First, find the prime factorization of 72: $72 = 2^3 \times 3^2$.
2. Since $n^2$ is divisible by $2^3 \times 3^2$, the prime factorization of $n^2$ must contain at least three 2s and two 3s.
3. Because $n^2$ is a perfect square, all exponents in its prime factorization must be even. Therefore, the minimum exponents for $n^2$ are $2^4$ and $3^2$.
4. If $n^2 = 2^4 \times 3^2$, then $n = \sqrt{2^4 \times 3^2} = 2^2 \times 3^1 = 4 \times 3 = 12$.
5. The smallest possible value of $n$ is 12.

2. Percent Change: A retail store increased the price of a jacket by 20%. After a month, the store discounted the new price by 25%. What was the total percentage change from the original price?

1. Let the original price be $100 for simplicity.
2. Increase by 20%: $100 + (0.20 \times 100) = 120$.
3. Decrease the new price by 25%: $120 - (0.25 \times 120) = 120 - 30 = 90$.
4. Calculate the total change: $\frac{90 - 100}{100} \times 100 = -10\%$.
5. The price decreased by 10%.

3. Ratios: In a bag of marbles, the ratio of red to blue marbles is 3:4, and the ratio of blue to green marbles is 8:5. What is the ratio of red to green marbles?

1. Identify the common element: Blue marbles.
2. The first ratio is $R:B = 3:4$. The second ratio is $B:G = 8:5$.
3. Multiply the first ratio by 2 to match the blue marble units: $R:B = 6:8$.
4. Now that the blue units are equal (8), the combined ratio is $R:B:G = 6:8:5$.
5. The ratio of red to green is 6:5.

Practice Questions

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

• $xy$
• $x + 2y$
• $2x + y$
• $x^y$

2. What is the remainder when $3^{15}$ is divided by 5?

3. A set contains seven consecutive integers. If the sum of the integers is 105, what is the largest integer in the set?

4. Quantity A: The number of distinct prime factors of 60.
Quantity B: The number of distinct prime factors of 90.

5. A machine produces 120 units in 3 hours. How many units will it produce in 7 hours at the same rate?

6. If 40% of $x$ is 20% of $y$, and $x = 30$, what is the value of $y$?

7. Which of the following is equivalent to $\frac{1}{4} + \frac{1}{5} + \frac{1}{10}$?

8. The average (arithmetic mean) of five numbers is 20. If a sixth number, 32, is added to the set, what is the new average?

9. If $n$ is a positive integer, what is the greatest common divisor of $4n$ and $6n$?

10. A mixture contains water and alcohol in a ratio of 5:2. If there are 14 liters of alcohol, how many total liters are in the mixture?

Answers & Explanations

1. Answer: $2x + y$
   Even + Even = Even; Even + Odd = Odd. $2x$ is always even regardless of $x$. Adding $y$ (an odd integer) to an even number always results in an odd number.
2. Answer: 2
   Look for a pattern in the units digit of powers of 3: $3^1=3, 3^2=9, 3^3=27, 3^4=81$. The units digits repeat every 4 powers: 3, 9, 7, 1. For $3^{15}$, divide 15 by 4 to get a remainder of 3. The units digit is the 3rd in the sequence, which is 7. When any number ending in 7 is divided by 5, the remainder is 2.
3. Answer: 18
   Let the middle integer be $x$. The sum is $(x-3)+(x-2)+(x-1)+x+(x+1)+(x+2)+(x+3) = 7x$. Set $7x = 105$, so $x = 15$. The largest integer is $x + 3 = 18$.
4. Answer: The two quantities are equal.
   Prime factors of 60: 2, 3, 5 (Total 3). Prime factors of 90: 2, 3, 5 (Total 3).
5. Answer: 280
   Rate = $\frac{120}{3} = 40$ units per hour. In 7 hours, $40 \times 7 = 280$.
6. Answer: 60
   Equation: $0.40(30) = 0.20(y)$. $12 = 0.20y$. $y = \frac{12}{0.20} = 60$.
7. Answer: $0.55$ or $\frac{11}{20}$
   Find a common denominator (20): $\frac{5}{20} + \frac{4}{20} + \frac{2}{20} = \frac{11}{20} = 0.55$.
8. Answer: 22
   Sum of five numbers = $20 \times 5 = 100$. New sum = $100 + 32 = 132$. New average = $\frac{132}{6} = 22$.
9. Answer: $2n$
   The factors of 4 are 1, 2, 4. The factors of 6 are 1, 2, 3, 6. The largest common factor is 2. Including the variable, the GCD is $2n$.
10. Answer: 49
    Ratio units for alcohol = 2. If $2 \text{ parts} = 14$, then $1 \text{ part} = 7$. Total parts = $5 + 2 = 7$. Total liters = $7 \times 7 = 49$.

1. If the price of an item is reduced by 10% and then increased by 10%, how does the final price compare to the original?

• AIt is the same.
• BIt is 1% lower.
• CIt is 1% higher.
• DIt is 10% lower.

Frequently Asked Questions

What arithmetic topics are most common on the GRE?

The exam frequently tests number properties (even/odd, divisibility), percent change, ratios, and basic operations with fractions or decimals. You should also be comfortable with the GRE Prep concepts of mean, median, and weighted averages.

Can I use a calculator for arithmetic questions on the GRE?

Yes, an on-screen calculator is provided for the Quantitative Reasoning section, but it is best used for complex multiplication or division. For many arithmetic properties, logical reasoning is faster than manual entry.

How do I handle remainders in GRE arithmetic?

Remainders are best understood through the formula $\text{Dividend} = ( \text{Divisor} \times \text{Quotient}) + \text{Remainder}$. For large exponents, identify cycles in units digits to find remainders when dividing by 5 or 10.

Is 1 considered a prime number on the GRE?

No, by definition, a prime number is an integer greater than 1 that has exactly two distinct positive divisors. Therefore, 2 is the smallest prime number and the only even prime.

How can I improve my speed on arithmetic problems?

Improving speed requires memorizing common fraction-to-decimal conversions and squares up to 20. Additionally, using tools like an AI Question Generator can provide the high-volume practice needed to recognize patterns instantly.