Proving $(a \mod n + b \mod n)\mod k = a \mod k + b \mod k$ if $k|n$

Question

I'm trying to show: $(a\mod n + b\mod n)\mod k= a\mod k+b\mod k$ if $k|n$

Got stuck at $((a\mod n)\mod k + (b\mod n)\mod k)\mod k=(a\mod k + b\mod k)\mod k $

I don't know how to eliminate the last $\mod k$. What am I missing?

Answer 1

Given $k|n$. Take left hand side of which can be reduced to $((a+b)\mod n)\mod k = ((a+b)\mod kt)\mod k$ for some t in integers. So clearly left hand side becomes $(a+b)\mod k$ which is right hand side.