2013 AMC 12B 第 9 题

Question

能整除

12!

的最大完全平方数，其平方根的质因数分解中各指数之和是多少？

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

分解

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

。

Factor

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

最大平方因数只保留偶数指数；再把指数减半得到平方根。

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

因为

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

，能整除它的最大完全平方数为

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

，其平方根为

2^5\cdot 3^2\cdot 5

。指数和为

5 + 2 + 1 = 8

。正确答案是 C。

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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