Balti Tee 2011 · Valikvooru ülesanne

Since $1001$ is divisible by $13$, so is $111 \times 1001 = 111111$. Noting that $2011 = 6 \times 334 + 7$, by taking off blocks of $111111$ from $N$ we deduce that $111X111$ is divisible by $13$.

Now reduce the number further by subtracting multiples of $1001$, obtaining multiples of $13$ at every step: $111X111 \rightarrow 11(X-1)111 \rightarrow 1(X-1)011 \rightarrow (X-1)001 \rightarrow (X-2)00 \text{ is divisible by } 13. \text{ So } X=2.$