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：质数奇偶性极限情形界定

难度评级：1370

解答：

在 $4$ 和 $18$ 之间的质数是 $5$ 、 $7$ 、 $11$ 、 $13$ 和 $17$ 。两个奇数的乘积是奇数、和是偶数，所以 $xy - (x + y)$ 是奇数。

又因为 $xy - (x + y)$ $= (x - 1)(y - 1) - 1$ 会随着任一质数增大而增大，结果从 $5 \cdot 7 - 12 = 23$ 到 $13 \cdot 17 - 30 = 191$ 。

选项中唯一在 $[23, 191]$ 内的奇数是 $119 = 11 \cdot 13 - (11 + 13)$ 。

所以正确答案是 C。

The primes between $4$ and $18$ are $5,$ $7,$ $11,$ $13,$ and $17.$ The product of two of them is odd and the sum is even, so $xy - (x + y)$ is odd.

Since $xy - (x + y)$ $= (x - 1)(y - 1) - 1$ increases as either prime increases, the result ranges from $5 \cdot 7 - 12 = 23$ up to $13 \cdot 17 - 30 = 191.$

The only odd option in $[23, 191]$ is $119 = 11 \cdot 13 - (11 + 13).$

Thus, the correct answer is C.

2000 AMC 10 第 11 题

答案：C

难度评级：1370

解答：

其他年份的第 11 题