I brute force the solution of order 5 in $Z^*_{36}$ by using the following $a^5 \equiv 1 \mod 36$ and I see that there is no solution for this. However, I don't quite know how to prove this. Can someone help me?
2026-04-12 12:36:02.1775997362
Find all element of order 5 in $Z^*_{36}$
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The group $\mathbb Z_{36}^*$ has order $12$. Therefore, the order of each of its elements must divide $12$.