Suppose $a,b,c,d$ to be whatever quantities whatsoever that satisfy the proportion $\frac{a}{b}=\frac{c}{d}$. Is there a value for $a$ other than a factor or a multiple of $c$. Or, is there a value for $b$ other than a factor or a multiple of $d$.
I am sure there cannot be such a value but at the same time there is no way for me to prove it.
The given expression is ad=bc if a and b are two different numbers such that their greatest common factor is 1
Case-1: If Z is least common multiple of a and c, b=z/c and d=z/a clearly satisfies the equation. ( you may change the variables)
Case-2: If cd=ad=0 then c may or may not be equal to a
Case-3: if above are not the cases then they has to be surely equal.