Define addition of a set of symbols {0, 1} by:

$$0 + 0 = 0, \ 0 + 1 = 1, \ 1 + 0 = 1, \ 1 + 1 = 0$$

Define multiplication by n where n is a non-negative integer by:

$$0·0=0, \ 0·1=0, \ n·0=0, \ n·1=1+1+···+1 \ for \ n>0$$ 

How can you prove that if $n$ is odd then $n·1=1$ and that if $n$ is even then $n·1=0$.

I am having a really hard time trying to understand this. 

I was thinking that I should sub. $n=2k+1$ because it is odd and try working from there. I was wondering if anyone could give me a hint or start me off.