another math problem (help!)

[Guest]

This was in my hw the other day and I was wondering if anyone could help...

I need to write a proof, and idk how to even start :(

4. Let x_1, x_2, . . . , x_19 be positive integers each of which is less than or equal to 93. Let y_1, y_2, . . . , y_93 be positive integers each of which is less than or equal to 19. Prove that there exists a (nonempty) sum of some x_i’s equal to a sum of some y_j’s.

[Patashu]

I suspect the answer has something to do with modulus arithmetic but beyond that I don't have a clue. It seems as if trying to generate a counterexample (e.g. all xs 93, all ys 19, adjust 93s until no multiple of 19 adds up to any of them) leads you into a trap whereby adding them in order won't work but out of order there's always some way to do it, some residue that is forced to be assembled