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2013 AMC 12B Problem 9

Problem 9 of 25EasierNumber TheoryCounting & Probability

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

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Solution

Since 12!=210⋅35⋅52⋅7⋅11,12! = 2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 11, the largest perfect square dividing it is 210⋅34⋅52,2^{10}\cdot 3^4\cdot 5^2, whose square root is 25⋅32⋅5.2^5\cdot 3^2\cdot 5. The exponents sum to 5+2+1=8.5 + 2 + 1 = 8. Thus, the correct answer is C.

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Concepts: prime factorization · factorial · perfect square

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.