Positive integers $m$ and $n$ satisfy $m n=5000$. If $m$ is not divisible by 10 and $n$ is not divisible by 10 , what is the value of $m+n$ ?
We note that $5000=5 \cdot 1000=5 \cdot 10^{3}=5 \cdot(2 \cdot 5)^{3}=5 \cdot 2^{3} \cdot 5^{3}=2^{3} 5^{4}$.

Since neither $m$ nor $n$ is divisible by 10 , then neither $m$ nor $n$ can have factors of both 2 and 5 . Since the product $m n$ equals 5000, the only prime factors of 5000 are 2 and 5 , and neither $m$ nor $n$ has factors of both 2 and 5 , then $m$ and $n$ must be $2^{3}$ and $5^{4}$, or $5^{4}$ and $2^{3}$.

Therefore, $m+n=2^{3}+5^{4}=8+625=633$.

ANSWER: 633