Let $R(q)$ be the number of real characters mod $q$. A character $\chi \mod q$ is called real if $\chi(a)\in\mathbb{R}$ for every $a\in \mathbb{Z}$, which means $\chi(a)\in\{-1,1\}$ for every $a\in\mathbb{Z}$ with gcd$(a,q)=1$.
I want to show that this $R$ is multiplicative, so $R(ab)=R(a)R(b)$ for $a,b\in\mathbb{Z}$ with gcd$(a,b)=1$. I'm trying to prove this with induced characters, but I'm not getting it completely. Can someone help me to understand it? Thanks!