Divisibility by 7

Theorem
An integer $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is divisible by $7$ if and only if $\displaystyle \sum_{i=0}^{n-1} (3^i X_i)$ is divisible by $7$.

Direct Proof
The first addend is always divisible by $7$ because $10^i - 3^i$ always produces a number divisible by $7$ from the difference of two powers. So $X$ will be divisible by $7$ if and only if the second addend is divisible by $7$.