AMC12 Number Theory 3

Question 1

What is Euler's totient function phi(12)?

Accepted Answer

4

Answer

phi(12)=12*(1-1/2)*(1-1/3)=4.

Question 2

How many solutions does x^2 = 1 (mod 8) have among {0,1,2,...,7}?

Accepted Answer

4

Answer

1^2=1, 3^2=9=1, 5^2=25=1, 7^2=49=1 (all mod 8), so there are 4 solutions.

Question 3

What is the largest prime factor of 2310?

Accepted Answer

11

Answer

2310=2*3*5*7*11, so the largest prime factor is 11.

Question 4

How many trailing zeros does 50! end with?

Accepted Answer

12

Answer

Trailing zeros equal the number of factors of 5: floor(50/5)+floor(50/25)=10+2=12.

Question 5

If n=2^3*3*5^2, how many ordered pairs (a,b) satisfy lcm(a,b)=n?

Accepted Answer

105

Answer

For each prime power p^k, pairs with max exponent k number (2k+1); total=(7)(3)(5)=105.

AMC12 - American Mathematics Competition 12 Practice Test