I saw the following on mathfail.com, and thought about it a bit

Facebook math fail

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

[WolframCDF source=”https://raw.githubusercontent.com/peeterjoot/mathematica/master/misc/sumOfdigits.cdf” width=”600″ height=”528″ altimage=”http://peeterjoot.com/wp-content/uploads/2014/07/Capture.png” altimagewidth=”536″ altimageheight=”363″]

However, I’m unsure how to prove or disprove this?