Intuition behind Order of an element of finite cyclic group.

Question

Let $\color{brown}{C}$ be a finite cyclic group, and $\color{brown}{a\in C}$ , such that $\color{brown}{|a|=n}$, then $$\color{green}{|a^k|=\frac{n}{gcd(k,n)}=\frac{lcm(k,n)}{k}}$$How to intuitively think/justify the above mathematical statement. $$\color{grey}{any\quad help\quad is \quad deeply\quad appreciated}$$