A Generalization of A Result of Gauss on Primitive Root

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $\mu(p-1)$ modulo $p$, where $\mu$ is the M\"{o}bius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.