Nilpotent elements in Zn

Question

Can you help me for my algebra homework? Let n be square free positive integer. Prove that Zn has no nonzero nilpotent element.

Answer 1

Hint:

Suppose $\;a\in\Bbb Z_n\;$ is a nonzero nilpotent element, then

$$\;a^m=0\pmod n\;\iff a^m=kn\;,\;\;k\in\Bbb Z$$

Now write $\;a\;$ as a product of primes (Fundamental Theorem of Arithmetic), use the above and get a contradiction...