Show that there are no integer solutions $m, n$ of the equation $$\left({5\over 4}\right)^m = \left({2\over 1}\right)^n$$ which gives the number of major thirds in an octave on a musical scale.
Given integers $a, b, c$ and $d$ where $a$ and $b$ are coprime and $c$ and $d$ are coprime, find necessary and sufficient conditions for there to exist positive integers $m$ and $n$ such that $$\left({a\over b}\right)^m = \left({c\over d}\right)^n.$$