$\newcommand{\lcm}{\operatorname{lcm}}$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 $x,y$?
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You ask for strict inequality: $\text{lcm}(x,y)<\text{lcm}(a,b)$. But even if $x<a$ and $y<b$ one can get equality, say for $x=2$, $y=3$, $a=6$ and $b=6$. – Angina Seng Apr 29 '20 at 08:32
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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
Bill Dubuque
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@Kishalay I clarified what was meant (it uses only your hypothesis). – Bill Dubuque Apr 29 '20 at 05:37