Prove $\,a\mid A,\, b\mid B\,\Rightarrow\, {\rm lcm}(a,b)\mid {\rm lcm}(A,B)$

Question

$\newcommand{\lcm}{\operatorname{lcm}}$Suppose $x\mid a$ and $y\mid b$ then can we say that $\lcm(x,y)<\lcm(a,b)$. I had tried to prove it but I am unable to do it. I am a little weak with number theory. Is the statement true? It would also be helpful if someone provided a graph of $\lcm$ function and help me to know how $\lcm$ varies with $x,y$?

Answer 1

Hint $\ \begin{align} x\mid a\mid{\rm lcm}(a,b)\\ y\mid b \mid{\rm lcm}(a,b)\end{align} \Rightarrow\, {\rm lcm}(x,y)\mid {\rm lcm}(a,b)$

using that: $\,\ x,y\mid m\iff {\rm lcm}(x,y)\mid m,\,$ the LCM Universal Property