2002 AMC 10B Problem 6

Below is the professionally curated solution for Problem 6 of the 2002 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10B solutions, or check the answer key.

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Concepts:factoring prime

Difficulty rating: 1070

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

none

one

two

more than two, but finitely many

infinitely many

Solution:

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.

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