If $m$ is a 3-digit positive integer such that $\mathop{\text{lcm}}[8m,10^{10}] = 4\cdot\mathop{\text{lcm}}[m,10^{10}]$, then what is the value of $m$?
I don't know what to do in this situation. Is there any identity or theorem I'm supposed to use? Help is greatly appreciated.
Looking at the condition strongly suggest that the prime $2$ will be the key here.
Let $a=v_2(m)$ be the order of $2$ dividing $m$. we remark that (in general) $$v_p(\text {lcm} (c,d))=\max(v_p(c),v_p(d))$$ So your condition implies that $$\max(a+3,10)=\max(a,10)+2$$
It is easily seen (by trial and error if nothing else) that this implies $a=9$. Thus $2^9=512$ divides $m$ but as $m$ has only got $3$ digits we must have $$\boxed{m=512}$$