Maximizing a base?

Note that $(n = 1) \mod{n-1}$ . As such, the test for divisibility by $n-1$ in base $n$ is to check if the sum of the digits is divisible by $n-1$ (whats the divisibility test for $9$ in base $10$ ?). Adding the digits:

$n-1+n-2+n-3+n-4+n-5 \mod n-1$

$5n-15 \mod n-1$

$-10 \mod n-1$

For the number to be divisible, $n-1$ must divide $-10$ . Therefore, $n-1$ can be equal to: $-10, -5, -2, -1, 1, 2, 5, or 10$ . Choosing the maximum value results in $n=11$