2013 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2013 AMC 12B, 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.

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Concepts:prime factorization factorial perfect square

Difficulty rating: 1510

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!\,?$

$5$

$7$

$8$

$10$

$12$

Solution:

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.

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