# Is there a game or analogy for searching for a number $R$ that is multiple of some number $A$ but not multiple of another $B$?

$$\DeclareMathOperator{\lcm}{lcm}$$There are many applications for finding the GCD or LCM of two numbers. I'm now interested in finding anything that could be used to illustrate the following mathematical relationship.

Let $$r = \lcm(r_1, r_2)$$. Find a divisor $$c$$ of $$r$$ such that $$r/c$$ is a multiple of $$r_1$$ but not a multiple of $$r_2$$ (or a multiple of $$r_2$$ but not a multiple of $$r_1$$). Notice there are cases in which no such $$c$$ can be found. For example, when $$r_1 = r_2 = \lcm(r_1, r_2)$$.

Is there any interesting motivation that [can] be put on this?

• This doesn't quite fit, but the question somehow reminds me of Conway's Sylver Coinage game. – Adam Feb 6 '20 at 14:59
• Not sure if this helps, but it's equivalent to asking for a set $C$ such that $C \subseteq B-A$. Such an $A$, $B$, and $C$ would satisfy $C \subseteq A \cup B$ (equivalent to $c \mid r$), $A \subseteq (A \cup B) - C$ (equivalent to $r_1 \mid r/c$), but $B \not\subseteq (A \cup B)-C$ (equivalent to $r_2 \nmid r/c$). – Aeryk Feb 6 '20 at 19:00
• @Aeryk, how is "$c$ divides $r$" equivalent to "$C \subseteq A \cup B$"? – user724963 Feb 7 '20 at 14:52
• Not sure if "analogous" or "equivalent" is the better choice of word. Either way, you consider a natural number to be a multiset of its prime factors and you get the translation(?) I described. – Aeryk Feb 7 '20 at 15:26
• @user724963 That is not the only case in which no $c$ can be found (though I notice now that I had a typo: I meant $r_1|r_2$, not the reverse). – user3482749 Feb 11 '20 at 9:39