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=(akak−1…a2a1a0)10=k∑j=0ak−j10k−j. The expression
Q′3(n)=(a2a1a0)10−(a5a4a3)10+(a8a7a6)10−…
are called alternating sum of the digits of third order of n. For example,
Q′3(123456789)=789−456+123=456
Proposition: 7|n ⇔ 7|Q′3(n).
proof. ??
Thanks for any help.
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