AMC 10 · 2007 · #23

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

difference-of-squaresfactorsparitydivisor-count caseworksystematic-enumeration ↑ Prerequisites: difference-of-squares

📏 Medium solution 💡 3 insights

Problem

Count the ordered pairs (m,n) of positive integers with m ≥ n whose squares differ by 96, that is, m² - n² = 96.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Chasing m² - n² = 96 by testing squares one at a time is slow. The unlocking move is Tool #4 (Introduce a Variable): a difference of squares always factors, m² - n² = (m-n)(m+n), so the condition becomes a product of two whole numbers equal to 96. Naming those two factors turns the problem into 'split 96 into a product of two pieces.' From there Tool #2 (Make a Systematic List) counts the factor pairs, and Tool #3 (Eliminate Possibilities) throws out the pairs that would force m or n to be a fraction.

1STEP 1

Factor the difference of squares

A difference of two squares always factors: m² - n² = (m-n)(m+n). So the condition m² - n² = 96 says m-n and m+n multiply to 96.

m² - n² = (m-n)(m+n) = 96

2STEP 2

Name the two factors

Let a = m - n and b = m + n, so a·b = 96. Reversing, m = (a+b)/2 and n = (b-a)/2, so each factor pair gives exactly one pair (m,n).

a = m-n, b = m+n, a · b = 96, m = (a+b)/2, n = (b-a)/2

3STEP 3

Both factors must be even

For m = (a+b)/2 to be whole, a and b need the same parity; their product 96 is even, so they cannot both be odd — both are even.

a+b even → a,b same parity; a · b = 96 even → not both odd → a,b both even

4STEP 4

Count the even factor pairs

With a = 2p, b = 2q, pq = 24 and p less than q: 1×24, 2×12, 3×8, 4×6 — 4 pairs, choice (B): (25,23), (14,10), (11,5), (10,2).

pq = 24, p ＜ q: (1,24),(2,12),(3,8),(4,6) → 4 pairs → (B)

Answer

Check each pair directly: 25² - 23² = 625 - 529 = 96, 14² - 10² = 196 - 100 = 96, 11² - 5² = 121 - 25 = 96, and 10² - 2² = 100 - 4 = 96. All four work, and no other even factor pair of 96 was missed, so the count of 4 is solid. It also makes sense that the answer is small: 96 has only a handful of even-times-even splittings.

💡Key takeaway

A difference of squares m² - n² equals (m-n)(m+n), so counting the pairs becomes counting the ways to split 96 into two even factors.

Factor the difference of squares
Name the two factors
Both factors must be even
Count the even factor pairs

Factor the difference of squares

Name the two factors

Both factors must be even

Count the even factor pairs

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