If $999\times \mathrm{abc}=\mathrm{def123}$ in decimal system, then find $\mathrm{a,b,c,d,e\ and\ f}.$ Consider the following multiplication in decimal system:
$$999\times \mathrm{abc}=\mathrm{def123}$$
then find the value of digits $\mathrm{a,b,c,d,e\ and\ f}.$
Here $\mathrm{abc}$ means not $(a\times b\times c$), $\mathrm{abc}$ is a number of 3 digits decimal system (e.g. if $\mathrm{abc}=123$, then $
\mathrm{a=1,\ b=2,\ c=3}$).

Using the expansion of a natural number into powers of $10$ your equation becomes
\begin{equation*}
999\times 10^2a+999\times 10b+999c=10^5d+10^4e+10^3f+123.\tag{1}
\end{equation*}
This implies that $c=7$ because the final digit of the right-hand side of $(1)$ is $3 $, so should be the left-hand side's; and since we have $9(7)=63$, the unit's digit of $999c $ shall equal $3 $. Plugging this value into $(1)$,  simplifying and dividing both sides by $10$, we obtain
\begin{equation*}
999\times 10a+999b+687=10^4d+10^3e+10^2f,\tag{2}
\end{equation*}
which yields $b=7$ because now the final digit of both sides of $(2)$ must equal $0$, and since $9(7)+7=70$, the final digit of $999b+687 $ shall equal $0 $. Proceeding in a similar way to the above, we find that the single solution of $(1)$ is 

$a=8$, $b=7$, $c=7$, $d=8$, $e=7$, $f=6$.

