Let p be a prime. Let r1, r2, ..., rp-1 be the integers 1, 2, ..., p-1 in some order. Prove that some two of the numbers
1*r1, 2*r2, ..., (p-1)*rp-1
must be congruent modulo p.
[Hint: If not, think of their product modulo p.]
(The 1, 2, and p-1 when following an r are supposed to be subscript. I can't seem to figure out how to make them and keep them that way.)