GRE Number Theory and Integer Properties

Question 1

What is the greatest common divisor (GCD) of 252 and 198?
(A) 6
(B) 9
(C) 18
(D) 36

Accepted Answer

18

Answer

Using the Euclidean algorithm: GCD(252, 198) → GCD(198, 54) → GCD(54, 36) → GCD(36, 18) → GCD(18, 0) = 18.

Question 2

What is the least common multiple (LCM) of 12, 18, and 24?
(A) 36
(B) 48
(C) 72
(D) 144

Accepted Answer

72

Answer

12 = 2²×3; 18 = 2×3²; 24 = 2³×3. LCM = 2³×3² = 8×9 = 72.

Question 3

How many prime numbers are between 50 and 70?
(A) 4
(B) 5
(C) 6
(D) 7

Accepted Answer

5

Answer

Primes between 50 and 70: 53, 59, 61, 67. That's 4 primes. Wait — also check 71 is not included. Actually: 53, 59, 61, 67 = 4 primes.

Question 4

If n is an integer, which of the following must be even?
(A) n² + 1
(B) n(n+1)
(C) 2n + 1
(D) n² − n + 1

Accepted Answer

n(n+1)

Answer

n(n+1) is always even because it is the product of two consecutive integers, one of which must be even.

Question 5

What is the remainder when 7^100 is divided by 5?
(A) 0
(B) 1
(C) 2
(D) 4

Accepted Answer

1

Answer

7 mod 5 = 2. Powers of 2 mod 5 cycle: 2,4,3,1,2,4,3,1... period 4. 100 mod 4 = 0, so 7^100 mod 5 = 2^100 mod 5 = 2^(4×25) mod 5 = (2^4)^25 mod 5 = 1^25 = 1.

GRE Practice Test

GRE Practice Test

GRE Number Theory and Integer Properties