Free practice questions/GMAT Focus Edition
GMAT Focus Edition — Number Properties
6 free practice questions with full explanations.
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Start freeQuestion 1
If n is a positive integer and 15n is divisible by 6, which of the following must be a factor of n?
- A) 2
- B) 3
- C) 5
- D) 9
Show answer & explanation
Correct answer: A) 2
15n divisible by 6 means it's divisible by both 2 and 3. Since 15 is already divisible by 3, that part is automatic. But 15 is odd, so for 15n to be divisible by 2, n itself must be even — meaning 2 must be a factor of n.
Question 2
When positive integer x is divided by 5, the remainder is 2. When x is divided by 3, the remainder is 1. What is the remainder when x is divided by 15?
- A) 2
- B) 4
- C) 7
- D) 11
Show answer & explanation
Correct answer: C) 7
We need x ≡ 2 (mod 5) and x ≡ 1 (mod 3). Testing x = 7: 7 divided by 5 leaves remainder 2, and 7 divided by 3 leaves remainder 1. Both conditions hold, and 7 < 15, so the remainder mod 15 is 7.
Question 3
How many positive integers less than 30 are prime?
- A) 8
- B) 9
- C) 10
- D) 11
Show answer & explanation
Correct answer: C) 10
The primes less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 — a total of 10 primes.
Question 4
The greatest common divisor of two positive integers is 8 and their least common multiple is 96. If one of the integers is 24, what is the other integer?
- A) 16
- B) 24
- C) 32
- D) 48
Show answer & explanation
Correct answer: C) 32
For any two positive integers, GCD x LCM = product of the two numbers. So 8 x 96 = 768, and the other integer is 768 / 24 = 32. Check: GCD(24, 32) = 8 and LCM(24, 32) = 96, confirming the answer.
Question 5
If a and b are integers and a x b is odd, which of the following must be true? I. a + b is even II. a is odd III. b is odd
- A) I only
- B) II only
- C) III only
- D) I, II, and III
Show answer & explanation
Correct answer: D) I, II, and III
A product of integers is odd only if every factor is odd. So both a and b must be odd (II and III are true). The sum of two odd numbers is always even, so a + b is even (I is true). All three statements must hold.
Question 6
How many positive divisors does 72 have?
- A) 8
- B) 9
- C) 10
- D) 12
Show answer & explanation
Correct answer: D) 12
72 = 2^3 x 3^2. The number of positive divisors is (3+1)(2+1) = 4 x 3 = 12.
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