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

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    $\begingroup$ Show that the generators preserve the partition into congruence classes mod $(k,n)$. $\endgroup$ – user10354138 Jun 2 at 21:05
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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.

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