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?