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Question 11

If $$m$$ and $$n$$ are positive integers such that $$\frac{m+n}{m^2+mn+n^2}=\frac{4}{49}$$, then $$m+n$$ is equal to

Cross multiplication gives $$49(m+n)=4(m^2+mn+n^2)$$. The positive integer solutions are $$\{m,n\}=\{6,10\}$$, which can be verified directly because $$49(16)=4(36+60+100)$$. Hence $$m+n=16$$.

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