AMC 10 · 2002 · #3

Grade 6 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
polynomial-factoringprime-numbersparity identify-subproblems ↑ Prerequisites: prime-numberspolynomial-factoring
📏 Medium solution 💡 2 insights
Problem
Look at the expression n² - 3n + 2. Feed in the positive integers n = 1, 2, 3, … one at a time and see how often the result is a prime number. The task is to count how many positive integers n make n² - 3n + 2 prime, and match that count to one of the five descriptions.

Pick an answer.

(A)
$\ \text{none}$
(B)
$\ \text{one}$
(C)
$\ \text{two}$
(D)
$\ \text{more\ than\ two,\ but\ finitely\ many}$
(E)
$\ \text{infinitely\ many}$
How to solve
Strategy Identify Subproblems

Plugging in numbers forever cannot prove a count of 'none,' 'one,' or 'infinitely many,' so raw checking is not enough by itself. Tool #7 (Identify Subproblems) supplies the key: the quadratic factors as (n-1)(n-2), which turns 'is this prime?' into a question about a product of two integers. A prime can only be a product of two whole numbers when one of them is 1, so the search collapses to a handful of cases. Tool #6 (Guess and Check) then tests just those few n values, and Tool #3 (Eliminate Possibilities) reads the final count off against the five choices.

1STEP 1

Factor the expression

The quadratic is a product in disguise: (n-1)(n-2), checked by expanding.

n² - 3n + 2 = (n-1)(n-2)
2STEP 2

Ask what makes a product prime

A prime product forces the smaller factor to be 1, so n-2 = 1.

(n-1)(n-2) prime → n-2 = 1 (smaller factor must be 1)
3STEP 3

Check the small cases

Testing small values, n=1 and n=2 give 0, while n=3 gives 2, which is prime.

n=1: 0, n=2: 0, n=3: 2×1 = 2 (prime)
4STEP 4

Rule out every larger n and count

For n at least 4 both factors exceed 1, so the product is composite — exactly one value works, choice (B).

n ≥ 4: (n-1)(n-2) composite → only n=3 → (B) one
Answer
one
Spot-check a few values directly against the original expression: n=3 gives 9-9+2 = 2 (prime), n=4 gives 16-12+2 = 6 = 2·3 (composite), n=5 gives 25-15+2 = 12 (composite), and n=1,2 give 0. Every check agrees with the factored analysis, and the values only grow past n=4, so nothing prime hides further out. Exactly one positive integer, n=3, produces a prime, confirming a count of one.
💡Key takeaway

Factor it into (n-1)(n-2): a product is prime only when one piece is 1, and that happens for just one positive n, namely n=3, giving the prime 2.

  • Factor the expression
  • Ask what makes a product prime
  • Check the small cases
  • Rule out every larger n and count

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