Question 8
(HUN 1) (a) Let $(m,k)=1$. Prove that there exist integers $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_k$ such that each product $a_ib_j$ ($i=1,2,\ldots,m$; $j=1,2,\ldots,k$) gives a different residue when divided by $mk$.
(b) Let $(m,k)\gt 1$. Prove that for any integers $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i,j)\ne(s,t)$) that give the same residue when divided by $mk$.
(b) Let $(m,k)\gt 1$. Prove that for any integers $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i,j)\ne(s,t)$) that give the same residue when divided by $mk$.