Let $a,b>0$ be coprime and not divisible by $2$ or $5$.
Now $$\frac1a=0.00\dots00r_100\dots00r_100\dots00r_1\dots$$ $$\frac1b=0.00\dots00r_200\dots00r_200\dots00r_2\dots$$
where $r_1$ and $r_2$ are repeating portions of fractions of lengths $\ell_1$ and $\ell_2$ respectively.
How to show that when we truncate both the sequences at length $\ell_1\ell_2$ and shift the sequences to the left above the decimal point and consider the resulting sequences as integers $m,n$ respectively then we have to have $b|m$ and $a|n$ and is it true $ab|mn$?
Example:
$\frac13=0.333333\dots$ and $\frac1{11}=0.09090909090909090909090909090909\dots$ and $\ell_1=1$ and $\ell_2=2$.
So truncate at $2$ significant positions and shifting left gives $m=33$ and $n=9$ (note there is no leading set of $0$s for $\frac1{11}$ and $r_2=09$. Now $11|m$ with $\frac m{11}=3$ and $3|n$ with $\frac n{3}=3$. $gcd(11,3)=1$ is consistent with what we have.