If $m$ and $n$ are integers and $\sqrt{mn}$ is $10.$ Which of the following cannot be the value of $m+n$?

The correct option is C: $50$

$\sqrt{mn}=10$

We know that $\sqrt{100}=10$

So, $mn=100$

If $m=20,$ then $n=\frac{100}{20}=5,m+n=20+5=25$

If $m=50,$ then $n=\frac{100}{50}=2,m+n=50+2=52$

If $m=100,$ then $n=\frac{100}{100}=1,m+n=100+1=101$

Hence, $D$ will be the correct option.