I'm having issues trying to simplify finding a "lower multiple". Essentially I'd like to be able to develop a short method to solve the following statement:
Assume you have two rational numbers, r1 and r2. You can also assume that
1)$$r1=\frac ab, \space r2=\frac cd,\space for\space some\space a, b, c, d \in \mathbb Z (where \space b,d \neq 0)$$ 2) $$ gcd(a,b) = gcd(c,d) = 1$$ and r1, r2 are simplified into their lowest terms
If r1 | r2, does b = d?
EDIT: After the multiplication has taken place, I don't want a' = c' and b' = d' after simplification.
BONUS: would b always be a multiple of d? (or vice versa). I wonder if the numerators sharing prime factors and denominators sharing prime factors has anything to do with it.
I've tried using my first/second year linear algebra proof methods but I can't seem to find a theorem that helps. I've drawn up a couple of examples and found it true, but I'm also biased because I probably want it to be true. If anyone knows a counterexample to this, could they please point it out?
Counter example: $$\frac{3}{8} | \frac{9}{4}$$