Show than $(1,k+1), (1,2,3,...,n)$ generate the group $S_n$ if and only if $k$ and $n$ are coprime

Question

I am able to prove constructively that if $(k,n)=1$ then we can generate $S_n$ but am struggling with the converse.

Answer 1

Let $n\geq 2$ and $1\leq a < b \leq n$. Then $S_n$ is generated by the transposition $(ab)$ and the $n$-cycle $(12\dots n)$ if and only if $\gcd(b-a,n)=1$.

For a complete proof see Theorem 2.8 of Keith Conrad's blurb on Generating Sets.