For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.



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> How big can $\mathsf{LCM}(\mathcal R(a,b))$ be if $a,b\in\big[\frac r2,r\big]$ hold and are coprime?
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On $\mathsf{LCM}$ of a set of integers
In other words, 
$$\mathcal R(a,b)=\{q\in[1,\min(a,b)]: q\mid(b-a)\}.$$
It is easy to see that for any $x,y\in \mathcal R(a,b)$, we have $\mathrm{LCM}(x,y)\mid(b-a)$, and therefore $\mathrm{LCM}(\mathcal R(a,b))\leq |b-a|$.


