2024 AMC 10A Problem 11

Because $m$ is a square root, $m \ge 0$. Squaring, $n^2 - 49 = m^2$, so
$$(n - m)(n + m) = 49.$$
The two factors are integers whose product is $49$: the ordered factor pairs are $(1, 49), (7, 7), (49, 1), (-1, -49), (-7, -7), (-49, -1)$. Solving $n - m = d$, $n + m = 49/d$ for each:

• $(1, 49)$: $n = 25, m = 24$. Valid.
• $(7, 7)$: $n = 7, m = 0$. Valid.
• $(49, 1)$: $n = 25, m = -24$. Rejected ($m < 0$).
• $(-1, -49)$: $n = -25, m = -24$. Rejected.
• $(-7, -7)$: $n = -7, m = 0$. Valid.
• $(-49, -1)$: $n = -25, m = 24$. Valid.

Check: $\sqrt{625 - 49} = \sqrt{576} = 24$ and $\sqrt{49 - 49} = 0$. Four ordered pairs: $(24, 25), (24, -25), (0, 7), (0, -7)$.

The answer is $\boxed{\textbf{(D)}\ 4}$.

An equation of the form $n^2 - m^2 = k$ is a difference of squares, and factoring turns "find integer solutions" into "list the factor pairs of $k$." The two hidden constraints, that $n$ can be negative and that $m$ cannot, are exactly what the answer choices are designed to test.

• Forgetting that n may be negative (answering 2), or allowing m = −24 even though m equals a square root and must be nonnegative.
• Missing the solutions with $m = 0$ from the factor pair $(7, 7)$ and $(-7, -7)$.