AMC12 2014 B

AMC12 2014 B · Q15

AMC12 2014 B · Q15. It mainly tests Exponents & radicals, Primes & prime factorization.

When $p = \sum_{k=1}^{6} k \ln k$, the number $e^{p}$ is an integer. What is the largest power of 2 that is a factor of $e^{p}$?

当$p = \sum_{k=1}^{6} k \ln k$时，数$e^{p}$是一个整数。$e^{p}$的因式中2的最大幂是多少？

(A) $2^{12}$ $2^{12}$

(B) $2^{14}$ $2^{14}$

(D) $2^{18}$ $2^{18}$

(E) $2^{20}$ $2^{20}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Because $k\ln k=\ln(k^k)$ and the log of a product is the sum of the logs, $p=\ln\prod_{k=1}^6 k^k$. Therefore $e^p$ is the integer $1^1\cdot2^2\cdot3^3\cdot4^4\cdot5^5\cdot6^6=2^{16}\cdot3^9\cdot5^5$, and the largest power of $2$ dividing $e^p$ is $2^{16}$.

答案（C）：因为 $k\ln k=\ln(k^k)$，且乘积的对数等于各因子对数之和，所以 $p=\ln\prod_{k=1}^6 k^k$。因此 $e^p$ 等于整数 $1^1\cdot2^2\cdot3^3\cdot4^4\cdot5^5\cdot6^6=2^{16}\cdot3^9\cdot5^5$，所以整除 $e^p$ 的最大的 $2$ 的幂为 $2^{16}$。

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