Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$.
I evaluated the inequalities into
$(ym-5)(xm-7) < 35$
$(yn-5)(xn-7) > 35$
but can't really think of what to do next.
Hint You need $m\le 12/n$ and $n\ge 12/m$ so $mn=12$, find the possibilities for positive integers $m<n$…