A $14$-digit. number $666666 XY 444444$ is a multiple of $26$. If $X$ and $Y$ are both positive, what is the smallest vaue of $X+ Y$? 
Overseas Competition->Knowledge Point->Number Theory Modules->Division without Remainders->Divisibility Rules
Since $1001$ is a multiple of $13$, $111111 = 111 \times 1001$ is also a multiple of $13$.  It follows that both $666666$ and $444444$ are both multiples of $26$.  $666666XY 444444 = 66666600000000 + XY 000000 + 444444$  $\Rightarrow XY$ must be divisible by $13$.  Smallest $X+Y=1+3=4$. 