Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also $a$ is a primitive root mod $p$. Show that $a$ is also a primitive root mod $2p$. Let $p$ be an odd prime. Suppose that $a$ is an odd integer and also
$a$ is a primitive root modulo $p$. Show that a is also a primitive root modulo
$2p$.
Any hints will be appreciated. Thanks very much.

A way is like this (you may have to justify the points a bit): 


*

*There are $p-1$ invertible classes mod $2p$, the same number as for $p$. 

*$a$ is invertible mod $2p$.

*The multiplicative order of $a$ mod $p$ is $p-1$. 

*The multiplicative order of $a$ mod $2p$ is at least as large as the one mod $p$. 
From this you can conclude directly.  
