AMC12 2013 B

AMC12 2013 B · Q9

AMC12 2013 B · Q9. It mainly tests Primes & prime factorization, Counting divisors.

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

求能整除 $12!$ 的最大的完全平方的平方根的质因数的指数之和。

(A) 5 5

(B) 7 7

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Because 12! = $2^{10}\cdot 3^{5}\cdot 5^{2}\cdot 7\cdot 11$, the largest perfect square that divides 12! is $2^{10}\cdot 3^{4}\cdot 5^{2}$ which has square root $2^{5}\cdot 3^{2}\cdot 5$. The sum of the exponents is $5+2+1=8$.

答案（C）：因为 $12!=2^{10}\cdot 3^{5}\cdot 5^{2}\cdot 7\cdot 11$，所以整除 $12!$ 的最大完全平方数是 $2^{10}\cdot 3^{4}\cdot 5^{2}$，其平方根为 $2^{5}\cdot 3^{2}\cdot 5$。指数之和为 $5+2+1=8$。

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