Question 23

The sum of all positive integers $$m, n$$ which satisfy $$m^2 + 2mn + n = 44$$ is

Correct Answer: 18

Solution

Solving for $$n$$ gives $$n = \frac{44 - m^2}{2m + 1}$$, which must be a positive integer, so $$m$$ can only be 1, 2, 3, 4, 5 or 6. Testing these, only $$m = 2$$ with $$n = 8$$ and $$m = 3$$ with $$n = 5$$ work. The sum of all these values is $$2 + 8 + 3 + 5 = 18$$.

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