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DETAILS OF THE FIFTH MOURA VELHO RULE

This post is designed to clarify details of the fifth Moura Velho rule of divisibility by 7 presented in a recent video as a bonus.Changing what must be changed, this rule works also to test divisibility by 13.

THE ALGORITHM

N = abc; xabc → 7|xa; S = x + bc; if 7|S then 7|N

For larger numbers, this algorithm must be applied repetitively to each period of N. The inverse additive mod 7 of each sum (S) must be added to the digit “x” of the next period and to the number formed by the two following digits.

The procedure must be applied until the rightmost period of N is reached. If 7|FR (final result) then 7|N. If 7ƗFR then FR mod 7 is the remainder of the division of N by 7.

If the initial period is incomplete, the algorithm must be applied partially.

WHY IT WORKS

Digit “x” is equivalente to 2a mod 7 and bc is equivalente to ( 3b + c ) mod 7. Then the sum of the products mod 7 obtained by the application of the algorithm is: SP = 2a + 3b + 1c and the multipliers are respectively: 2, 3 and 1, exactly the same multipliers established by Pascal in his criterion for divisibility by 7 (theorem 2.5) to the three digits of the period of the first order, as mentioned before.

Applying the inverse additive mod 7 to SP:

─ ( 2a + 3b + c ) mod 7 ≡ 5a + 4b + 6c.

For larger numbers, the following period is formed by the digits “def”. The application of the algorithm to the following period results in SP = 2d + 3e + f and the aggregated sum of both periods is SP = 5a + 4b + 6c + 2d + 3e + f, whose multipliers are: 5, 4, 6, 2, 3 and 1 that are the multipliers established by Pascal when he created his criterion for divisibility by 7.

If the number is formed by various periods, the successive and cumulative application of the inverse additive mod 7 to each sum obtained in the passage of one period to another implies the successive repetition of the multipliers established by Pascal: ...31546231546231

The application of this rule is valid because it is equivalent to the application in mod 7 of the multipliers established by Pascal in his criterion for divisibility by 7.

If 7|FR (final result) then 7|N; if 7ƗFR then FR mod 7 is equivalent to the remainder of the division of N by 7.

HOW IT WORKS AND REMAINDER

Mental calculation may be performed very quickly. Notes are used only to illustrate the application of the rule.

N = 62,324,452; 62; (6)324, (1)452
─ 62 mod 7 + 6 + 24 = 31; ─ 31 mod 7 + 1 + 52 = 57; 7Ɨ57 and 7ƗN; RF = 57; 57 mod 7 ≡ 1 = remainder of the division of N by 7.

N = 362,458,923,312; (6)362, (1)458, (4)923, (6)312
6 + 62 = 68; ─ 68 mod 7 + 1 + 58 = 61;

─ 61 mod 7 + 4 + 23 = 29; ─ 29 mod 7 + 6 + 12 = 30; 7Ɨ30 and 7ƗN; RF = 30; 30 mod 7 ≡ 2 = remainder of the division of N by 7.