Question

能整除 $13^4 - 11^4$ 的最大的 $2$ 的幂是多少？

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

$8$

$16$

$32$

$64$

$128$

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使用平方差分解：

$\begin{aligned} 13^4-11^4 &=(13^2+11^2)\\ &\qquad\cdot(13^2-11^2)\\ &=290\cdot48\\ &=32\cdot435. \end{aligned}$

因为 $435$ 是奇数，所以 $32$ 是能整除该表达式的最大的 $2$ 的幂。

所以正确答案是 C。

We can factor this expression using difference of squares.

$\begin{aligned} 13^4-11^4 &=(13^2+11^2)\\ &\qquad\cdot(13^2-11^2)\\ &=290\cdot48\\ &=32\cdot435. \end{aligned}$

Because $435$ is odd, $32$ is the largest power of $2$ that divides the expression.

Thus, C is the correct answer.