I know that $S_n$ is generated by a number of things, like all transpositions, all transpositions of form $(1a)$, the transpositions $(12),(23),(34),\cdots(n-1n)$, and just the two elements $(123\cdots n),(12)$.
Suppose $n$ is prime. If you just have $(123\cdots n)$ and some arbitrary transposition $(ab)$, how does this also generate $S_n$?
Can you somehow get to $(12)$ or reduce it to some other previous case?