Let $M$ be a monoid, $x \in M$ and assume $$x^k = x^{k+d}$$ holds for some $k \geq 0$ and $d>0$. Assume that also $$x^{k'} = x^{k' + d'}$$ holds for some $k' \geq 0$ and $d'>0$. Does this imply that $$x^{\min(k,k')} = x^{\min(k,k') + \mathrm{gcd}(d,d')}?$$
Equivalently: Let $\sim$ be a congruence on the monoid $(\mathbb{N},+)$ with $k \sim k+d$ and $k' \sim k' + d'$. Does this imply $\min(k,k') \sim \min(k,k') + \mathrm{gcd}(d,d')$?
Correct me if I am wrong but in any finite cyclic monoid $\langle x \rangle = C_{n,m}$ your statement is always true. It follows that if $x^k = x^{k+d}$ then $k\geq n$ and we have $x^k = x^{k+d} \Leftrightarrow d = 0\pmod m$.