Determine all positive integers $r$ such that there exist at least two pairs of positive integers $(m, n)$ satisfying the equation $$2^m = n! + r.$$

I have found that for $r=2$, we have $m=2,n=2$ and $m=3,n=3$ and for $r=8$ we have $m=5,n=4$ and $m=7,n=5$

Further, $r$ can not be odd. How we can prove in general. Please give some idea.