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A primitive root modulo an integer $n$ is the generator of the multiplicativegroup 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 theM\"{o}bius function. In this paper, we will generalize these two interestingcongruences and give the congruences of the sum and the product of integerswith the same index modulo $n$.

A Generalization of A Result of Gauss on Primitive Root