## How do I prove the following divisibility rule of?

This is an exercise in modular arithmetics. First note that 1000\equiv -1\mod 7 . According to modular arithmetics \begin{align*}x &\equiv a\mod n, y\equiv b\mod n \Rightarrow\\ xy&\equiv ab \mod n\end{align*} and x + y \equiv a+b\mod n So \sum\limits_{k=0}^N 10^{3k} \alpha_k=\sum\limits_{k=0}^N 1000^{k} \alpha_k\equiv \sum\limits_{k=0}^N (-1)^k \alpha_k\mod 7 In your case N=2,\alpha_0=851,\alpha_1=369,\alpha_2=1.