So, I know that c=aq and c=br for some integers q and r. I also know and can prove that c is greater than or equal to d, thus c=dz + t for some integers z and t by Euclid's algorithm. We know that t is greater than or equal to zero and less than d. If we can show d|t, then we know t=0. From here, we have c=dz, thus d|c and by substitution, [a,b]|c and we're done.
The part I don't get is this. How do you show d|t? If I knew how to do this, I could do the rest of the problem.