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If $n \geq 2$ is an integer and $a, b > 0$ are integers such that $$(2a + b) \mid (a+b)^n,$$ what relations necessarily follow?

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Since $a+b\equiv -a\pmod{2a+b}$, we find $a^n\equiv 0\pmod{2a+b}$. Especially, any prime $p$ dividing $2a+b$ must also divide $a$ (and hence also $b$)

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let $a+b=s$ and $a=t$. What are the conditions for $s+t | t^n$?

Well clearly $s$ and $t$ can't be relatively prime, but that's about all we can say about them.

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