I've been trying to justify the following step in a textbook (Iwaniec-Kowalski) proof, but just can't seem to:

$\displaystyle \displaystyle \sum_{n\:\equiv\: 0 \mod k} \chi (n) = \sum_{l|k} \mu (l) \sum_{(n,l)=1} \chi (n)$

where $\displaystyle \mu$ is the Mobius function. It's clearly some application of Mobius inversion. Can anyone show this rigorously?