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

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2007 AMC 10A Problem 23

Below is the professionally curated solution for Problem 23 of the 2007 AMC 10A, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:difference of squares Diophantine Equation parity

Difficulty rating: 1400

23.

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

$3$

$4$

$6$

$9$

$12$

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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