Help with modular arithmetic

If$r_1,r_2,r_3,r_4,\ldots,r_{ϕ(a)}$ are the distinct positive integers less than $a$ and coprime to $a$, is there some way to easily calculate, $$\prod_{k=1}^{\phi(a)}ord_{a}(r_k)$$

Answer

The claim is true, with the stronger condition that there is some $i$ with $e_i=1$ and all other exponents are zero. The set of $r_i$'s is called a reduced residue system.

The second (now deleted) claim is false. Let $a=7$. Then $2^13^1=6^1$, two different representations.