Divisibility rules/Rule 1 for 7 proof

Proof

This is based on divisibility by 11 rule

Assume $N$ has $3k$ digits, otherwise add zeros to the left. WLOG let $N = a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6a_5a_4a_3a_2a_1a_0$ where the $a_i$ are base-ten numbers. Then $N = 10^{3k-1} a_{3k-1} + 10^{3k-2} a_{3k-2} + 10^{3k-3} a_{3k-3} \cdots + 10^8 a_8 + 10^7 a_7 + 10^6 a_6 + 10^5 a_5 + 10^4 a_4 + 10^3 a_3 + 10^2 a_2 + 10 a_1 + a_0$ $= 1000^{k-1} (100 a_{3k-1} + 10 a_{3k-2} + a_{3k-3}) \cdots + 1000^2 (100 a_8 + 10 a_7 + a_6) + 1000 (100 a_5 + 10 a_4 + a_3) + (100 a_2 + 10 a_1 + a_0).$

Rewriting or partitioning $N$ into 3 digit numbers ( $a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6 a_5a_4a_3 a_2a_1a_0$ ). $N = 1000^{k-1} (a_{3k-1}a_{3k-2}a_{3k-3}) \cdots + 1000^2 (a_8a_7a_6) + 1000 (a_5a_4a_3) + (a_2a_1a_0).\\ \equiv (-1)^{k-1} (a_{3k-1}a_{3k-2}a_{3k-3}) \cdots + (-1)^2 (a_8a_7a_6) - (a_5a_4a_3) + (a_2a_1a_0) \pmod {7}, \text{since} 1000\equiv -1\pmod{7}$

This is the alternating sum of groups of 3 digit numbers of $N$ , which is what we wanted.

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Divisibility rules/Rule 1 for 7 proof

Proof

See also