Does connecting edges $m$ distance apart in a cycle $C_n$ result in a new cycle?

Question

Given a simple cycle $C_n$ with red edges, prove that connecting all vertices $m$ distance apart with a blue edge results in a blue simple cycle that spans all vertices of $C_n$ if and only if $\gcd(n,m)=1$.

Is there a way to characterize this in terms of something other than graph theory?

Answer 1

It is a result in number theory. Vertex $m$ ends up in a cycle containing {$m,2m,3m,\dots, km, \dots\}$ (where every number is reduced $\bmod n$.)

So in other words, the graph you obtain consists of various dijoint cycles. Exactly $(m,n)$ cycles, each with $n/(n,m)$ vertices.