If $o(x) = k$ we surely have that: $x^k = e$, therefore:

$(Nx)^k = N(x^k) = Ne = N$, the identity of $G/N$. Hence $o(Nx)$ must divide $k$, for if not, say: $o(Nx) = m$ with:

$k = qm + r$ and $0 < r < m$, we have:

$N = (Nx)^k = (Nx)^{qm+r} = (Nx)^{qm}(Nx)^r = ((Nx)^m)^q(Nx)^r = N^q(Nx)^r = N(Nx)^r = (Nx)^r$

contradicting the fact that $m$ is the least positive integer with $(Nx)^m = N$.