AMC12 Number Theory

Question 1

What is the remainder when 7^100 is divided by 5?

Accepted Answer

1

Answer

7=2(mod5); powers of 2 mod 5 cycle as 2,4,3,1 with period 4; since 100=0(mod4), remainder is 1.

Question 2

How many positive divisors does 360 have?

Accepted Answer

24

Answer

360=2^3*3^2*5, so the number of divisors is (3+1)(2+1)(1+1)=24.

Question 3

What is gcd(48, 36)?

Accepted Answer

12

Answer

48=36*1+12 and 36=12*3+0, so gcd=12.

Question 4

What is the smallest positive integer with exactly 6 divisors?

Accepted Answer

12

Answer

12=2^2*3 has (2+1)(1+1)=6 divisors, and no smaller positive integer has exactly 6.

Question 5

How many integers from 1 to 100 are divisible by 3 or 5?

Accepted Answer

47

Answer

By inclusion-exclusion: 33+20-6=47.

AMC12 - American Mathematics Competition 12 Practice Test