(a) Let
![\gcd(m, k) = 1](/media/m/7/2/1/721966e13c146911157f54a181500e49.png)
. Prove that there exist integers
![a_1, a_2, . . . , a_m](/media/m/5/d/b/5db87f8e19ae77b431349d8d68e15eea.png)
and
![b_1, b_2, . . . , b_k](/media/m/9/3/3/9337b87eb9a007f9201591c07040fd92.png)
such that each product
![a_ib_j](/media/m/6/4/9/6493d4b844817b71f6c5b5912660e1c0.png)
(
![i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k](/media/m/c/b/f/cbf343e30bbcbfa381fa9ac0ee799089.png)
) gives a different residue when divided by
![mk.](/media/m/c/2/b/c2be2bb245e4cf2970154fa65d8e750a.png)
(b) Let
![\gcd(m, k) > 1](/media/m/8/d/f/8df0e9339bc080c78e4d608464846fa8.png)
. Prove that for any integers
![a_1, a_2, . . . , a_m](/media/m/5/d/b/5db87f8e19ae77b431349d8d68e15eea.png)
and
![b_1, b_2, . . . , b_k](/media/m/9/3/3/9337b87eb9a007f9201591c07040fd92.png)
there must be two products
![a_ib_j](/media/m/6/4/9/6493d4b844817b71f6c5b5912660e1c0.png)
and
![a_sb_t](/media/m/a/a/e/aaeaa284ecde430f7c1db46aeff62a41.png)
(
![(i, j) \neq (s, t)](/media/m/7/e/a/7ea56b15768165a5f9104ab0b701a8a1.png)
) that give the same residue when divided by
![mk.](/media/m/c/2/b/c2be2bb245e4cf2970154fa65d8e750a.png)
Proposed by Hungary.
%V0
(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.