AMC12 Number Theory 2 — Questions and Answers
Question 1: What is the sum of all positive divisors of 28?
- 28
- 42
- 14
- 56 (Correct answer)
Correct answer: 56
28=2^2*7; sum of divisors=(1+2+4)(1+7)=7*8=56.
Question 2: How many two-digit prime numbers exist?
- 15
- 25
- 18
- 21 (Correct answer)
Correct answer: 21
There are 21 primes between 11 and 97: 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.
Question 3: What is lcm(12, 18)?
- 6
- 72
- 216
- 36 (Correct answer)
Correct answer: 36
lcm(12,18)=12*18/gcd(12,18)=216/6=36.
Question 4: For which prime p is p^2+2 also prime?
- 2
- 5
- 7
- 3 (Correct answer)
Correct answer: 3
p=3 gives 9+2=11 (prime); all primes p>3 give p^2=1(mod3) making p^2+2=0(mod3).
Question 5: What is the units digit of 7^53?
- 3
- 9
- 1
- 7 (Correct answer)
Correct answer: 7
Units digits of powers of 7 cycle as 7,9,3,1 with period 4; 53=1(mod4), so units digit is 7.
Question 6: How many perfect squares are strictly between 100 and 400?
- 8
- 10
- 12
- 9 (Correct answer)
Correct answer: 9
From 11^2=121 to 19^2=361, there are 19-11+1=9 perfect squares.
AMC12 Number Theory 2
What is the sum of all positive divisors of 28?