jmc

We find that 1000 gives a quotient of 90 and a remainder of 10 when divided by 11. Therefore, if we subtract 10 from 1000 we should subtract 10 from the remainder as well, which gives us zero. This means that $1000-10=\boxed{990}$ is divisible by 11, while the next number divisible by 11 is $990+11=1001.$