I know that given two numbers $a$ and $b$ such that $a=bq+r$ then $(a,r)$ and $(a,b)$ are the same. But I want to know if this is true with the LCM as well. I think this is untrue, because if we let the GCD be equal to $k$, then $\frac{a}{k}$ and $\frac{b}{k}$ must be coprime by the definition of GCD. Then, we know that the LCM of a and b is different to the LCM of a and r because they have different prime factors in them.
Is my reasoning correct? If it is not, what is the correct proof? Please help me.