GRE Number Theory and Integer Properties 2

Question 1

If n is a positive integer, what is the remainder when 7^100 is divided by 5?

Accepted Answer

1

Answer

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?

Accepted Answer

12

Answer

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?

Accepted Answer

42

Answer

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?

Accepted Answer

6

Answer

If k ≡ 3 (mod 7), then 2k ≡ 6 (mod 7), giving remainder 6.

Question 5

Which of the following is a prime number?

Accepted Answer

83

Answer

83 has no divisors other than 1 and itself, while 91=7×13, 87=3×29, and 85=5×17.

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