a quick math challenge

[Thomas Klemm]

(12-31-2013 03:03 PM)Curlytop Wrote:  Can anybody explain why this works for 7 (and 13)?

\[ \begin{align}
1000&\equiv&6 (7) \\
&\equiv&-1 (7) \\
\end{align} \]
\[ \begin{align}
1000&\equiv&12 (13)\\
&\equiv&-1(13) \\
\end{align} \]
\[ \begin{align}
123,456,789&=&123\times1000^2+456\times1000+789 \\
&\equiv&123\times(-1)^2+456\times(-1)+789 (7) \\
&\equiv&123-456+789 (7) \\
\end{align} \]

Cheers
Thomas