AMC12 Number Theory 1 — Questions and Answers
Question 1: What is the remainder when 7^100 is divided by 5?
- 4
- 2
- 3
- 1 (Correct answer)
Correct answer: 1
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?
- 18
- 12
- 36
- 24 (Correct answer)
Correct answer: 24
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)?
- 6
- 4
- 24
- 12 (Correct answer)
Correct answer: 12
48=36*1+12 and 36=12*3+0, so gcd=12.
Question 4: What is the smallest positive integer with exactly 6 divisors?
- 18
- 6
- 24
- 12 (Correct answer)
Correct answer: 12
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?
- 33
- 20
- 53
- 47 (Correct answer)
Correct answer: 47
By inclusion-exclusion: 33+20-6=47.
Question 6: What is 13^2 mod 7?
- 6
- 2
- 4
- 1 (Correct answer)
Correct answer: 1
13=-1(mod7), so 13^2=(-1)^2=1(mod7).
AMC12 Number Theory 1
What is the remainder when 7^100 is divided by 5?