Question

对多少个正整数 $n$ ，表达式 $n^2 - 3n + 2$ 是质数？

For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?

没有

none

一个

one

两个

two

多于两个，但有限个

more than two, but finitely many

无穷多个

infinitely many

答案：B

知识点：因式分解质数

难度评级：1070

解答：

分解得 $n^2 - 3n + 2 = (n-1)(n-2)$ 。

当 $n \ge 4$ 时，两个因数都大于 $1$ ，乘积为合数。当 $n = 1$ 或 $n = 2$ 时，原式为 $0$ ；当 $n = 3$ 时，原式为 $(2)(1) = 2$ ，是质数。

所以恰好有一个 $n$ 可行。

所以正确答案是 B。

Factor as $n^2 - 3n + 2 = (n-1)(n-2).$

For $n \ge 4,$ both factors exceed $1,$ so the product is composite. For $n = 1$ and $n = 2$ the value is $0,$ and for $n = 3$ the value is $(2)(1) = 2,$ which is prime.

So exactly one value of $n$ works.

Thus, the correct answer is B.

2002 AMC 10B 第 6 题

答案：B

难度评级：1070

解答：

其他年份的第 6 题