2024 DMC Mock 10 Problems/Problem 13

Let the integer we are trying to find be $x$ .

Then, $x$ satisfies $\frac{48}{x^2}+x=7$ . Since $x$ and $7$ are both integers, $\frac{48}{x^2}$ must also be an integer. $x$ must be $\pm 1,2,4$ . Guess and check yields $x=4$ . $4$ has $\boxed{3}$ factors.

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2024 DMC Mock 10 Problems/Problem 13