A simple application of PHP. All cubes are congruent to $0,1,-1,8,or -8$ modulo 13 i.e. at most $5$ diff. remainders. By PHP, there are at least $4$ such cubes leaving the same remainder modulo $13$. Now, any cube is congruent to $0,1 or -1$ modulo $7$. Again by PHP, there are at least two among these $4$ cubes which leave the same remainder modulo $13$. Hence, the diff. of these two cubes is divisible by $13*7=91$ as required.

What is PHP?

PigeonHole Principle