AMC8 2016

AMC8 2016 · Q15

AMC8 2016 · Q15. It mainly tests Factoring, Primes & prime factorization.

What is the largest power of 2 that is a divisor of $13^4 - 11^4$?

$13^4 - 11^4$的最大2的幂次除数是多少？

(A) 8 8

(B) 16 16

(D) 64 64

(E) 128 128

Answer

Correct choice: (C)

正确答案：（C）

Solution

First, we use difference of squares on $13^4 - 11^4 = (13^2)^2 - (11^2)^2$ to get $13^4 - 11^4 = (13^2 + 11^2)(13^2 - 11^2)$. Using difference of squares again and simplifying, we get $(169 + 121)(13+11)(13-11) = 290 \cdot 24 \cdot 2 = (2\cdot 8 \cdot 2) \cdot (3 \cdot 145)$. Realizing that we don't need the right-hand side because it doesn't contain any factor of 2, we see that the greatest power of $2$ that is a divisor $13^4 - 11^4$ is $\boxed{\textbf{(C)}\ 32}$.

首先，使用平方差公式$13^4 - 11^4 = (13^2)^2 - (11^2)^2 = (13^2 + 11^2)(13^2 - 11^2)$。再次使用平方差并化简，得$(169 + 121)(13+11)(13-11) = 290 \cdot 24 \cdot 2 = (2\cdot 8 \cdot 2) \cdot (3 \cdot 145)$。右侧不含2的因子，因此$13^4 - 11^4$的最大2的幂次除数为$\boxed{\textbf{(C)}\ 32}$。

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