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**Zalren** Show that if $\displaystyle m$ is a number having primitive roots, then the product of the positive integers less than or equal to $\displaystyle m$ and relatively prime to $\displaystyle m$ is congruent to $\displaystyle -1 (\mod{m})$.

$\displaystyle m$ has primitive roots means there is a least residue $\displaystyle a$ such that $\displaystyle a^{\phi(m)} \equiv 1 (\mod{m})$

Since $\displaystyle a$ is a primitive root of $\displaystyle m$, then the least residues $\displaystyle (\mod{m})$ of $\displaystyle a, a^2, ..., a^{\phi(m)}$ are a permutation of the $\displaystyle \phi(m)$ positive integers less than $\displaystyle m$ and relatively prime to $\displaystyle m$.

So the congruency to look at is $\displaystyle (a * a^2 * ... * a^{\phi(m)}) \equiv -1 (\mod{m})$ correct??? I have been looking at this congruency and failing to make it true.

Thanks for any help.