Question

从 $4$ 和 $18$ 之间选出两个不同的质数。用它们的乘积减去它们的和，可能得到下列哪个数？

Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$21$

$60$

$119$

$180$

$231$

答案：C

知识点：质数奇偶性因式分解

难度评级：1310

解答：

$4$ 和 $18$ 之间的质数是 $5, 7, 11, 13$ ，和 $17$ 。

对这样的两个质数， $xy - (x + y)$ $= (x - 1)(y - 1) - 1$ ，是两个偶数的乘积减去 $1$ ，因此为奇数。选项中只有 $119$ 是奇数。 $3 \pmod 4$ 。 $231$ 。 $(x-1)(y-1)=232$ ， $\{4,6,10,12,16\}$ $232$ 。

实际上， $\begin{aligned} 11 \cdot 13 - (11 + 13) &= 143 - 24 \\ &= 119. \end{aligned}$

因此，正确答案是 C。

The primes between $4$ and $18$ are $5, 7, 11, 13,$ and $17.$

For two such primes, $xy - (x + y)$ $= (x - 1)(y - 1) - 1$ is a product of two even numbers minus $1,$ hence it is $3 \pmod 4.$ This leaves $119$ and $231.$ The latter would require $(x-1)(y-1)=232,$ but no two distinct numbers in $\{4,6,10,12,16\}$ have product $232.$

Indeed, $\begin{aligned} 11 \cdot 13 - (11 + 13) &= 143 - 24 \\ &= 119. \end{aligned}$

Thus, the correct answer is C.

2000 AMC 12 第 6 题

答案：C

难度评级：1310

解答：

其他年份的第 6 题