AMC 10 · 2002 · #6

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)

$\ \text{infinitely\ many}$

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Two numbers multiplying to 2 and adding to -3 are -1 and -2, so n² - 3n + 2 factors as (n-1)(n-2).

n² - 3n + 2 = (n-1)(n-2)

2STEP 2

Ask what makes a product prime

A prime splits only as 1 times itself, so the smaller factor n-2 must equal 1; if both factors pass 1 the product is composite.

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

3STEP 3

Check the small cases

n = 1 and n = 2 both give 0, but n = 3 gives 2 · 1 = 2, a prime — exactly the case n-2 = 1.

n=1: 0, n=2: 0, n=3: 2×1 = 2 (prime)

4STEP 4

Rule out every larger n and count

For n ≥ 4 both factors are at least 2, so the product is composite; only n = 3 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

Factor the expression

Ask what makes a product prime

Check the small cases

Rule out every larger n and count

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