@n-dimen This is a bit late but for what it's worth, MSRI sent me a rejection letter last year. So they will likely send you one eventually if you don't get in
First off, you should note that the author doesn't mean to write $m>r$ and $s>1$. What he means is that $m>r>1$ and $m>s>1$.
Now, there is nothing fancy going on here. If $d=\gcd(x,m)$, then $d$ divides both $x$ and $m$, so we can write $m=rd$ and $x=sd$ for some integers $r,s$. Now, since...
Don't take the ignored emails too personally; this is totally standard for academics to ignore proposals they're not interested in, rather than directly deny the student. I'm assuming from this post that you're at a somewhat highly ranked school as well. You're in an unfortunate situation...