What is the largest M?

From Fermat we know that the largest $M$ such that $n^N-n$ is divisible by $M$ for all positive $n$ (where $N>1)$ is $M=\prod_{(p-1)|(N-1)}p$ , where $p$ is prime. In our case, the divisors of $N-1=10$ are 1,2,5 and 10, and the possible values for $p$ are 2, 3 and 11 (we reject 6 as it fails to be prime). Thus $M=2\times 3\times 11=\boxed{66}$ .