Index of Nilpotence for modular groups

Question

Why does $\mathbb{Z_{108}}$ have no elements of index nilpotence 4?

I know $r^4 = 0$, and the nilpotent elements must have factors of $3$ and $2$ in them but I can't seem to solve it?

Answer 1

We know that any nilpotent elements will need to look like $2^j3^k$ for positive integers $j$ and $k$ such that $j<2$ or $k<3$. Notice that, for any such integer (call it $x$), $108$ divides $x^3$. Or in other words, $x^3 \equiv 0 \pmod{108}$. So $4$ cannot be the least positive integer $k$ such that $x^k \equiv 0 \pmod{108}$.