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Let $a, b, c, d, r, s \in \mathbb{N}$. Find the necessary and sufficient conditions under which $r \mid (a-b)$ and $s \mid (c-d)$ $\implies$ $\operatorname{lcm}$ $(r,s)\mid(ac-bd)$.

A little thought is enough to find out some necessary conditions. But the sufficient requirement is what is making the problem difficult for me.

Actually the above problem is not the original problem that I have encountered, the original problem is,

Let $a, b, c, r, s \in \mathbb{N}$ such that $\operatorname{gcd}$ $(a,c)$ $=$ $1$. Find the necessary and sufficient conditions under which $r \mid (a-b)$ and $s \mid (c-1)$ $\implies$ $rs \mid(ac-b)$.

Any suggestion (or solution to any one of the problems) will be appreciated and a solution to any one of the problems may be posted without giving the solution to the other problem.

William Hilbert
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  • What necessary conditions do you seek to prove sufficient? It's difficult to help you with that if we don't know what they are. – Bill Dubuque Jul 06 '14 at 15:38
  • Question edited. Notice that if $r \mid s$ or $s \mid r$ then the claim obviously holds. But when this doesn't hold even then we find it to be true in some cases and false in other. So naturally, my question is that under which conditions we can always say that $\text {lcm}(r,s) \mid (ac-bd)$. – William Hilbert Jul 07 '14 at 13:35

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