Recall that a group is solvable if we can find a series of groups such that is normal in and is abelian.
Suppose that a group is a group of order , where and are distinct primes (and without loss of generality, suppose . By the first Sylow theorem, there are subgroups of of order . By the third Sylow theorem, the number of of these subgroups divides and satisfies . We can conclude that . Take to be that subgroup. By the second Sylow theorem, is normal in . We can write the series , noting that is abelian since it is prime order and is abelian since it is prime order .
This proves it.