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Prove that a_i and a_j exist for every prime p

Source:

October 5, 2010

modular arithmeticnumber theory proposednumber theory

Problem Statement

Let

p

be a prime number and

a_1, a_2, \ldots, a_{(p+1)/2}

different natural numbers less than or equal to

p.

Prove that for each natural number

r

less than or equal to

p

, there exist two numbers (perhaps equal)

a_i

and

a_j

such that

p \equiv a_i a_j \pmod r.