__Hint 1__: put \(\displaystyle \theta(1+n \mathbb{Z})=r+m\mathbb{Z}\) and extend \(\displaystyle \theta\) linearly to all \(\displaystyle \mathbb{Z}/n\mathbb{Z}\). now look at the conditions that will make \(\displaystyle \theta\) well-defined and multiplicative.

__Hint 2__: using the above, show that \(\displaystyle \theta\) is defined by \(\displaystyle \theta(x+n\mathbb{Z})=rx + m\mathbb{Z},\) where \(\displaystyle r\) satisfies the following conditions: \(\displaystyle \frac{m}{\gcd(n,m)} \mid r\) and \(\displaystyle m \mid r^2-r.\)