Tuesday, July 9, 2019

elementary number theory - Divisibility by 7 rule, and Congruence Arithmetic Laws

I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it.
Let n=(akak1a2a1a0)10=kj=0akj10kj. The expression
Q3(n)=(a2a1a0)10(a5a4a3)10+(a8a7a6)10


are called alternating sum of the digits of third order of n. For example,
Q3(123456789)=789456+123=456

Proposition: 7|n  7|Q3(n).




proof. ??



Thanks for any help.

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