Question

$2^{202} +202$ 除以 $2^{101}+2^{51}+1$ 的余数是多少？

What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1?$

$100$

$101$

$200$

$201$

$202$

答案：D

知识点：平方差模运算

难度评级：1880

视频讲解：

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文字解答：

令 $m=2^{101}+2^{51}+1$ 。围绕除数分解：利用平方差，这是 $m$ 的倍数。因此 $\begin{aligned} &2^{202}+202 \\ &\quad =(2^{101}+1)^2-(2^{51})^2+201. \end{aligned}$ $\begin{aligned} &(2^{101}+1)^2-(2^{51})^2 \\ &\quad =(2^{101}+2^{51}+1) \\ &\quad {}\cdot(2^{101}-2^{51}+1), \end{aligned}$ $2^{202}+202\equiv 201\pmod m.$

所以正确答案是 D。

Let $m=2^{101}+2^{51}+1.$ We factor the numerator around this divisor: $\begin{aligned} &2^{202}+202 \\ &\quad =(2^{101}+1)^2-(2^{51})^2+201. \end{aligned}$ By the difference of squares, $\begin{aligned} &(2^{101}+1)^2-(2^{51})^2 \\ &\quad =(2^{101}+2^{51}+1) \\ &\quad {}\cdot(2^{101}-2^{51}+1), \end{aligned}$ which is a multiple of $m.$ Therefore $2^{202}+202\equiv 201\pmod m.$

Thus, the correct answer is D .

2020 AMC 10B 第 22 题

答案：D

难度评级：1880

视频讲解：

文字解答：

其他年份的第 22 题