Baltic Way 2011 · Shortlist problem

Number Theory

A number $N$ , written in decimal notation, consists of $2011$ digits. All the digits are $1$ , except the middle digit. If $N$ is divisible by $13$ , find the middle digit.

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Topics

Divisibility and factorization

Solutions

Solution

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: $111X111 \rightarrow 11(X-1)111 \rightarrow 1(X-1)011 \rightarrow (X-1)001 \rightarrow (X-2)00 \text{ is divisible by } 13. \text{ So } X=2.$