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(a) Let \gcd(m, k) = 1. Prove that there exist integers a_1, a_2, . . . , a_m and b_1, b_2, . . . , b_k such that each product a_ib_j (i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k) gives a different residue when divided by mk.

(b) Let \gcd(m, k) > 1. Prove that for any integers a_1, a_2, . . . , a_m and b_1, b_2, . . . , b_k there must be two products a_ib_j and a_sb_t ((i, j) \neq (s, t)) that give the same residue when divided by mk.

Proposed by Hungary.

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