GRE Number Theory and Integer Properties 2 — Questions and Answers
Question 1: If n is a positive integer, what is the remainder when 7^100 is divided by 5?
- 1 (Correct answer)
- 2
- 3
- 4
Correct answer: 1
Powers of 7 mod 5 cycle 2,4,3,1 with period 4, and 100 is divisible by 4, giving remainder 1.
Question 2: How many positive divisors does 72 have?
- 6
- 8
- 10
- 12 (Correct answer)
Correct answer: 12
Since 72 = 2^3 × 3^2, the number of divisors is (3+1)(2+1) = 12.
Question 3: What is the greatest common divisor (GCD) of 84 and 126?
- 14
- 21
- 42 (Correct answer)
- 63
Correct answer: 42
84 = 2²·3·7 and 126 = 2·3²·7, so the GCD is 2·3·7 = 42.
Question 4: If the integer k leaves remainder 3 when divided by 7, what is the remainder when 2k is divided by 7?
- 3
- 5
- 6 (Correct answer)
- 1
Correct answer: 6
If k ≡ 3 (mod 7), then 2k ≡ 6 (mod 7), giving remainder 6.
Question 5: Which of the following is a prime number?
- 91
- 87
- 83 (Correct answer)
- 85
Correct answer: 83
83 has no divisors other than 1 and itself, while 91=7×13, 87=3×29, and 85=5×17.
Question 6: What is the least common multiple (LCM) of 12 and 18?
- 6
- 24
- 36 (Correct answer)
- 72
Correct answer: 36
12 = 2²·3 and 18 = 2·3², so the LCM is 2²·3² = 36.
Question 7: If n is an even integer, which expression must always be odd?
- n + 2
- n² + 1 (Correct answer)
- 2n + 4
- n/2
Correct answer: n² + 1
An even number squared is even, so n² + 1 is always odd.
GRE Number Theory and Integer Properties 2
If n is a positive integer, what is the remainder when 7^100 is divided by 5?