1. Multiplicative inverse properties

Hello,
I just have a question regarding multiplicative inverses.

Let $a \equiv p_1 \times p_2 \times \cdots \times p_n \pmod{x}$ where $p_i$ is a prime and $x$ is any positive integer and $\gcd(x, a) = 1$. Does the following hold :

$a^{-1} \equiv p_1^{-1} \times p_2^{-1} \times \cdots \times p_n^{-1} \pmod{x}$

Thank you

2. Yes! The group $(\mathbb{Z}/x\mathbb{Z})^\times$ of invertible residue classes $\mod x$ is an abelian group, and $a=a_1a_2 \Rightarrow a= a_1^{-1}a_2^{-1}$. The same holds for $a=a_1\dots a_n$.

3. Originally Posted by Bruno J.
Yes! The group $(\mathbb{Z}/x\mathbb{Z})^\times$ of invertible residue classes $\mod x$ is an abelian group, and $a=a_1a_2 \Rightarrow a= a_1^{-1}a_2^{-1}$. The same holds for $a=a_1\dots a_n$.
Thanks Bruno!