AMC12 Number Theory 2

Question 1

What is the sum of all positive divisors of 28?

Accepted Answer

56

Answer

28=2^2*7; sum of divisors=(1+2+4)(1+7)=7*8=56.

Question 2

How many two-digit prime numbers exist?

Accepted Answer

21

Answer

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)?

Accepted Answer

36

Answer

lcm(12,18)=12*18/gcd(12,18)=216/6=36.

Question 4

For which prime p is p^2+2 also prime?

Accepted Answer

3

Answer

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?

Accepted Answer

7

Answer

Units digits of powers of 7 cycle as 7,9,3,1 with period 4; 53=1(mod4), so units digit is 7.

AMC12 - American Mathematics Competition 12 Practice Test