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$$Question: What is the largest possible value of $\gcd(a,b)$, if the sum of two positive integers $a$ and $b$ is $1000$ and both $a$ and $b$ are divisible by a perfect square greater than $1$?
Let $d = \gcd(a,b)$. Then we can write $a = dx$, $b = dy$, where $\gcd(x,y) = 1$. Since $a + b = 1000$, we have:
dx + dy = d(x + y) = 1000 \Rightarrow d \mid 1000

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