Practice Arithmetic questions for the Graduate Record Examinations. Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Question 1easy
If n is an even integer, select all that apply — which of the following expressions must also be even?
- A.n + 2✓
- B.n × 3✓
- C.n + 1
- D.n²✓
- E.n + 7
Correct answer: A, B, D
An even integer added to another even integer remains even, so n + 2 is even (A). An even integer multiplied by any integer remains even, so n × 3 is even (B). n squared is even × even, which is even (D). Adding an odd number to an even number yields an odd number, so n + 1 and n + 7 are both odd (C and E). The core rule: even ± odd = odd, and even × anything = even.
Question 2easy
Select all that apply — which of the following integers are divisible by both 2 and 3?
Correct answer: A, C, E
A number divisible by both 2 and 3 must be divisible by 6. Divisibility by 2 requires an even last digit; divisibility by 3 requires the sum of digits to be divisible by 3. 18: even, 1+8=9 ✓. 24: even, 2+4=6 ✓. 36: even, 3+6=9 ✓. 22 is even but 2+2=4, not divisible by 3. 15 has digit sum 6 so divisible by 3, but it is odd so not divisible by 2.
Question 3easy
Select all that apply — which of the following fractions are greater than 1/2?
- A.3/5✓
- B.4/9
- C.5/8✓
- D.2/5
- E.7/12✓
Correct answer: A, C, E
A fraction is greater than 1/2 when the numerator is more than half the denominator. 3/5: 3 > 2.5 ✓ (equals 0.6). 5/8: 5 > 4 ✓ (equals 0.625). 7/12: 7 > 6 ✓ (≈0.583). 4/9: 4 < 4.5, so less than 1/2 (≈0.444). 2/5: 2 < 2.5, so less than 1/2 (equals 0.4). A quick cross-multiplication check — compare the fraction to 1/2 by checking if 2 × numerator > denominator.
Question 4easy
Select all that apply — which of the following expressions are equal to 40% of 150?
- A.60✓
- B.0.4 × 150✓
- C.(40/100) × 150✓
- D.150 ÷ 4
Correct answer: A, B, C
40% of 150 = (40/100) × 150 = 0.4 × 150 = 60. Options A, B, and C all represent the same value of 60 through equivalent expressions: A is the numerical result, B converts the percent to a decimal multiplier, and C uses the fraction form of 40%. Option D equals 37.5, which corresponds to 25% of 150, not 40%. The distractor D targets the error of confusing 40% with 1/4 (25%).
Question 5easy
Select all that apply — which of the following values of x satisfy the inequality |x| < 4?
- A.x = −5
- B.x = −3✓
- C.x = 0✓
- D.x = 4
- E.x = 2✓
Correct answer: B, C, E
|x| < 4 means the distance from x to 0 on the number line is less than 4, which is equivalent to −4 < x < 4. x = −3: |−3| = 3 < 4 ✓. x = 0: |0| = 0 < 4 ✓. x = 2: |2| = 2 < 4 ✓. x = −5: |−5| = 5, which is not less than 4. x = 4: |4| = 4, which is not strictly less than 4 (the inequality is strict). Option D is a common trap — students sometimes include the boundary value when the inequality is strict.
Question 6easy
Select all that apply. Which of the following numbers are prime?
Correct answer: B, C, E
A prime number has exactly two distinct positive factors: 1 and itself. The number 1 has only one factor (itself), so it is not prime by definition. 51 = 3 × 17, so it has more than two factors and is composite. 13, 29, and 37 each have no divisors other than 1 and themselves, confirming they are prime. A common error is to include 1 as prime or to overlook that 51 is divisible by 3 (since 5 + 1 = 6, a multiple of 3).
Question 7easy
Select all that apply. Which of the following are common factors of 24 and 36?
Correct answer: A, B, D
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors — numbers appearing in both lists — are 1, 2, 3, 4, 6, and 12, making A (4), B (6), and D (12) all correct. Option C (8) is a factor of 24 but not of 36, since 36 ÷ 8 = 4.5, which is not an integer. Note that 12 is the GCD of 24 and 36, and all common factors must themselves be factors of the GCD.
Question 8easy
Select all that apply. Which of the following fractions are equal to terminating decimals?
- A.1/4✓
- B.1/3
- C.3/8✓
- D.5/6
- E.7/20✓
Correct answer: A, C, E
A fraction in lowest terms produces a terminating decimal if and only if its denominator has no prime factors other than 2 and 5. 1/4: denominator 4 = 2², gives 0.25 ✓. 3/8: denominator 8 = 2³, gives 0.375 ✓. 7/20: denominator 20 = 2² × 5, gives 0.35 ✓. In contrast, 1/3 has a denominator with a factor of 3, giving the repeating decimal 0.333…, and 5/6 has a denominator with a factor of 3, giving 0.8333…. Recognizing which prime factors cause repeating decimals is the key reasoning skill here.
Question 9easy
The ratio of boys to girls in a class is 2:3. Select all that apply. Which of the following could be the total number of students in the class?
Correct answer: A, B, D
A ratio of 2:3 means students come in groups of 2 + 3 = 5, so the total must be a multiple of 5. Checking each option: 10 = 2 × 5 ✓ (4 boys, 6 girls); 15 = 3 × 5 ✓ (6 boys, 9 girls); 22 is not a multiple of 5 ✗; 25 = 5 × 5 ✓ (10 boys, 15 girls). A common mistake is choosing 22 because it seems like a reasonable class size, without checking whether it is consistent with the ratio. Any valid total must be divisible by the sum of the ratio's parts.
Question 10easy
A car travels 180 miles in 3 hours at a constant speed. Select all that apply. Which of the following statements about the car's travel are true?
- A.The car's speed is 60 miles per hour.✓
- B.The car travels 1 mile per minute.✓
- C.At this speed, the car would travel 240 miles in 4 hours.✓
- D.At this speed, the car would travel 100 miles in 2 hours.
Correct answer: A, B, C
The unit rate is 180 ÷ 3 = 60 miles per hour, confirming A. Converting to minutes: 60 miles per hour ÷ 60 minutes = 1 mile per minute, confirming B. At 60 mph for 4 hours: 60 × 4 = 240 miles, confirming C. For D, 60 mph × 2 hours = 120 miles, not 100 miles, so D is false. A common error is choosing D by misreading 'half the total time gives half the total distance from 180,' incorrectly computing 180 ÷ 2 = 90 rather than applying the unit rate correctly.