Divisibility rules/Rule 2 for 7 proof

Truncate the last digit of $N$, double that digit, and subtract it from the rest of the number (or vice-versa). $N$ is divisible by 7 if and only if the result is divisible by 7.


An understanding of basic modular arithmetic is necessary for this proof.

The divisibility rule would be $2d_0-k$, where $k=d_110^0+d_210^1+d_310^2+...$, where $d_{n-1}$ is the nth digit from the right (NOT the left) and we have $k-2d_0\equiv 2d_0+6k$ and since 2 is relatively prime to 7, $2d_0+6k\equiv d_0+3k\pmod{7}$. Then yet again $d_0+3k\equiv d_0+10k\pmod{7}$, and this is equivalent to our original number.

See also

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