# Math Help - Order of integers and primitive roots

1. ## Order of integers and primitive roots

Let a,m be in Z with m>0. If a' is the inverse of a modulo m, prove that the order of a modulo m is equal to the order of a' modulo m. Deduce that if r is a primitive root modulo m, then r' is a primitive root modulo m.

2. For the first part, note that $(a^{-1})^n=(a^n)^{-1}$ for all $n$.

3. Originally Posted by meshel88
Let a,m be in Z with m>0. If a' is the inverse of a modulo m, prove that the order of a modulo m is equal to the order of a' modulo m. Deduce that if r is a primitive root modulo m, then r' is a primitive root modulo m.
The first part as Tinyboss noted follows easily from consider elements of $\left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}$