AMC 12 Number Theory & Mathematical Logic 2

Question 1

What is the remainder when 2^100 is divided by 7?

Accepted Answer

2

Answer

Since 2^3 ≡ 1 (mod 7) and 100 = 3·33 + 1, 2^100 ≡ 2^1 = 2 (mod 7).

Question 2

How many positive divisors does 360 have?

Accepted Answer

24

Answer

360 = 2^3·3^2·5, so it has (3+1)(2+1)(1+1) = 24 divisors.

Question 3

What is gcd(2^12 − 1, 2^18 − 1)?

Accepted Answer

63

Answer

gcd(2^a − 1, 2^b − 1) = 2^gcd(a,b) − 1 = 2^6 − 1 = 63.

Question 4

What are the last two digits of 7^2026?

Accepted Answer

49

Answer

7^4 = 2401 ≡ 1 (mod 100) and 2026 ≡ 2 (mod 4), so 7^2026 ≡ 7^2 = 49 (mod 100).

Question 5

What is the contrapositive of "If n^2 is even, then n is even"?

Accepted Answer

If n is odd, then n^2 is odd.

Answer

The contrapositive of P → Q is ¬Q → ¬P, giving "if n is not even, then n^2 is not even."

AMC 12 - - American Mathematics Contest 12 Practice Test