2007 AMC 10A Problem 23 — Solution | LIVE by Po-Shen Loh

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2007 AMC 10A Problem 23

Attempt Problem 23 of the 2007 AMC 10A 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 2007 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

How many ordered pairs $(m, n)$ of positive integers, with $m \gt n,$ have the property that their squares differ by $96?$

$3$

$4$

$6$

$9$

$12$

Answer: B

Concepts:difference of squares Diophantine Equation parity

Difficulty rating: 1400

Solution:

Since $(m + n)(m - n) = 96$ and $96$ is even, both factors must be even.

The even factor pairs are $(48, 2),$ $(24, 4),$ $(16, 6),$ and $(12, 8),$ giving $(m, n) = (25, 23),$ $(14, 10),$ $(11, 5),$ and $(10, 2).$

So there are $4$ ordered pairs.

Thus, the correct answer is B.

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