jmc

First, factor the difference of squares.$$(m+n)(m-n)$$Since $m$ and $n$ are odd numbers, let $m=2a+1$ and $n=2b+1$, where $a$ and $b$ can be any integer.$$(2a+2b+2)(2a-2b)$$Factor the resulting expression.$$4(a+b+1)(a-b)$$If $a$ and $b$ are both even, then $a-b$ is even. If $a$ and $b$ are both odd, then $a-b$ is even as well. If $a$ is odd and $b$ is even (or vise versa), then $a+b+1$ is even. Therefore, in all cases, $8$ can be divided into all numbers with the form $m^2-n^2$. This can be confirmed by setting $m=3$ and $n=1$, making $m^2-n^2=9-1=8$. Since $8$ is not a multiple of $3$ and is less than $16$, we can confirm that the answer is $\boxed{8}$.