Exercise 16. Determine the number of integers divisible by 11, whose decimal representation is of the form $N=a b c d a b c d \cdots a b c d$, the pattern $a b c d$ being repeated 2016 times, and $a, b, c, d$ being digits such that $a \neq 0$.
Solution to exercise $16 \quad N=\overline{a b c d} \times 100010001 \cdots 0001$ is divisible by 11 if and only if $\overline{a b c d}$ is. We are therefore looking for the number of multiples of 11 among $1001,1012, \ldots, 9999$. There are $\frac{9999-1001}{11}+1=819$.