2016 AMC 8 第 15 题

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

将

13^4-11^4

分解为平方差

Factor

13^4-11^4

as a difference of squares.

分解后只统计乘积中

2

的幂

After factoring, count only the powers of

2

in the product.

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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}\text{。}

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.

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