Did you actually have a solution worked out? Or were you just frustrated and want to give up and peek? Here's a really big hint. See if that helps any.
There are 2049 natural numbers \(\le 2048\). So there are 2049 numbers of the form \(13^m\). However, there are only 2013 remainders when you divide numbers by 2013. Notice that 2013 is smaller than 2049.
Ok, so how about representing each value \(13^m\) using its quotient and remainder when divided by 2013. That is \(13^m = q_m2013 + r_m\).
How long since you had a meal? Too long. Go get food. Is it 2am. Go to bed. You can't do math when you are hungry or tired, especially this kind of problem.
If it's 2am before the Tuesday examlet, GO TO BED. Margaret said in lecture that this kind of evil trick question won't be on the examlets. Wait until after the examlet and give it another try.
Once you've tried all that, and either have a solution or are really stuck permanently, then look at the solution.