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

(D) more than two, but finitely many

2002 AMC 10B Problem 6

Attempt Problem 6 of the 2002 AMC 10B below, then check your answer against the professionally curated solution 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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For how many positive integers

n

n^2 - 3n + 2

a prime number?

none

one

two

more than two, but finitely many

infinitely many

Factor

n^2 - 3n + 2

For

(n-1)(n-2)

to be prime, one factor must equal

1

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.