Showing that in a given set of 7 distinct integers that there must be at least one pair in the set whose sum or difference is a multiple of 10.

Question

Show that, given any set of seven distinct integers, there must be at least one pair in the set whose sum or difference is a multiple of 10.

I've been thinking a lot on how to show that this is true, but I've not managed to come up with anything yet. If anyone could offer their viewpoint I'd appreciate it.

Answer 1

Modulo $10$, numbers are $0, (1,-1), (2,-2), (3,-3), (4,-4), 5$.

If six numbers are all distinct modulo $10$

• seventh is either identical to one of previous in which case difference is divisible by $10$.
• seventh is negative to one of previous, then sum is divisible by $10$.