smc

Substitute the second equation into the first one, we have $13x+11mx-11=700$. So $(13+11m)x=711$. So $13+11m$ is a factor of $711$. $711=3\ast 3\ast 79$, so the factors of $711$ are: $1,3,9,79,237,711$. Clearly, because $m\ge 1$, so $13+11m\ge 24$. So we only need to check whether $m$ is an integer when $11m+13=79,237,711$. When $11m+13=79$, $m=6$. Checking the other two choices, $m$ dosen't yield to be an integer. So $m=6$ is the only option. Select $\boxed{\text{6 only}}$.