#integer equations

Sum of digits of small powers of nine.

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In a previous post I wondered how to prove that for integer \(d \in [1,N]\)

\begin{equation}\label{eqn:numberGame:20} ((N-1) d) \text{mod} N + ((N-1) d) \text{div} N = N-1. \end{equation}

Here’s a proof in two steps. First for \(N = 10\), and then by search and replace for arbitrary \(N\).

\paragraph{\(N = 10\)}

Let

\begin{equation}\label{eqn:numberG…