Ireland_2017

To find the smallest number, we try $A=1$. The given equation becomes $1BCD=BCD+1BC\times D$, hence $1000=1BC\times D$. This means that $D$ is a divisor of $1000=2^3\times 5^3$ and so $D\in \{1,2,4,5,8\}$, because $D$ is a digit. On the other hand, $1BC<200$ implies that $1000<200\times D$, and so $D>5$. Therefore we need to have $D=8$ to find a solution. Because $1000/8=125$ we now see that there is a solution with $B=2$ and $C=5$. Hence the smallest possible value is $1258$.