If ad = bc, (a, b) = 1, (c, d) = 1, and a, b, c, and d are all positive, show a = c and b = d.

Question

So far I have that

Since we known that (a,b) = 1 by definition we can write it as ax+by=1. We also know that (c,d) = 1, so cx+dy=1. Since they are both equal to 1 I can set them equal to each other where ax+by=cx+dy.

Now I am unsure what my next steps would be to prove this. It seems that it would only work if the gcd of every variable is 1. But again, I am unsure how to show that.

Answer 1

Hint: $a\mid ad$, so $a\mid bc$, so $a\mid c$ because...

See if you can provide the reason for the last step and then complete the proof.