2013 AMC 12B Problem 9

Attempt Problem 9 of the 2013 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides

12!

5

7

8

10

12

Factor

12! = 2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 11

The largest square divisor keeps only even exponents; halving them gives the square root

Since

12! = 2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 11,

the largest perfect square dividing it is

2^{10}\cdot 3^4\cdot 5^2,

whose square root is

2^5\cdot 3^2\cdot 5.

The exponents sum to

5 + 2 + 1 = 8.

Thus, the correct answer is C.

Problem 9 in other years