Question

$16,384$ 的最大质因数是 $2$ ，因为 $16,384 = 2^{14}$ 。求 $16,383$ 的最大质因数的各位数字之和。

The greatest prime number that is a divisor of $16,384$ is $2$ because $16,384 = 2^{14}.$ What is the sum of the digits of the greatest prime number that is a divisor of $16,383?$

$3$

$7$

$10$

$16$

$22$

答案：C

知识点：质因数分解平方差

难度评级：1030

解答：

有因为 $129=3\cdot 43$ ，所以因此 $127$ 是最大质因数，其各位数字之和为 $10$ 。 $\begin{aligned}16,383 &= 16384-1 \\&= 2^{14}-1 \\&= (2^7-1)(2^7+1) \\&= 127\cdot 129.\end{aligned}$ $16383 = 3\cdot 43\cdot 127.$

所以答案是 C。

We know $\begin{aligned}16,383 &= 16384-1 \\&= 2^{14}-1 \\&= (2^7-1)(2^7+1) \\&= 127\cdot 129.\end{aligned}$ Since $129=3\cdot 43,$ we get $16383 = 3\cdot 43\cdot 127.$ Therefore, $127$ is the largest prime factor, and the sum of its digits is $10.$

Thus, the answer is C .

2021 AMC 10B Fall 第 8 题

答案：C

难度评级：1030

解答：

其他年份的第 8 题