I saw the following on mathfail.com (EDIT: dead link), and thought about it a bit
Notice that the +4 here is entirely misdirection. This is really just a statement that the sum of the digits of any integer power of 9 up to 81, is 9.
It also appears to be true that, for integer a in \([1, N+1]\)
\[(N a) \text{div} (N+1) + (N a) \text{mod} (N + 1) = N.\]
This is demonstrated in the following Mathematica Manipulate
However, I’m unsure how to prove or disprove this?