2002 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2002 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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Concepts:factoring prime

Difficulty rating: 1120

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 $n^2-3n+2=(n-1)(n-2).$ For $n\ge4$ both factors exceed $1,$ so the value is composite. Checking $n=1,2,3$ gives $0,$ $0,$ and $2.$ Only $n=3$ yields a prime, so there is exactly one such $n.$

Thus, the correct answer is B.

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