DETAILS OF THE FIRST MOURA VELHO RULE

This post is designed to clarify details of the first Moura Velho Rule of divisibility by 7 presented in a recent vídeo.

Generally, the Moura Velho rules work quickly through simple and successive mental calculations and dispense the use of pencil and paper.

Changing what must be changed, this rule works also to test divisibility by 11 and 13.

THE ALGORITHM

 N = a,bcd; a’ ≡ ( ─ cd mod 7 + a ) mod 7; cd is eliminated, resulting in a two-digit number a’b; if 7|a’b then 7|N.

This rule is applied to the pairs of digits of N from right to left.

WHY IT WORKS

─ cd mod 7 ≡ 6cd; 6cd is added to the place value of the thousands, resulting in an addition of 6,000 cd; as cd is eliminated, 1 cd is subtracted and 6,000 cd ─ cd = 5,999 cd.

As 7|5,999 the procedure preserves the value of N in mod 7.

 Observe that, as cd is eliminated (subtracted), two zero digits must replace cd.

HOW IT WORKS

N = 8,561; ( ─ 61 mod 7 + 8 ) mod 7 ≡ 3 → 35; 7|35 and 7|N

For larger numbers the algorithm must be repeated with the dislocation of a,bcd from right to left until the leftmost digit is reached. Eventually the last pair to the left may be incomplete; in this case, the digit “a” equals zero.

N = 69,218,683; ( ─ 83 mod 7 + 8 ) mod 7 ≡ 2; → 692126;

 ( ─ 26 mod 7 + 2 ) mod 7 ≡ 4; → 6941;

( ─ 41 mod 7 + 6 ) mod 7 ≡ 0; → 09; 7Ɨ9 and 7ƗN

THE REMAINDER

Let n = number of pairs of digits of N, including as a pair the eventual incomplete pair to the left.

Let FR = final result of the procedure.

If ( n ─ 1 ) mod 3 ≡ 0 then the remainder is FR mod 7.

If ( n ─ 1 ) mod 3 ≡ 1 then the remainder is 2 . FR mod 7.

If ( n  ─ 1 ) mod 3 ≡ 2 then the remainder is 4 . FR mod 7.

For N = 69,218,683; n = 4 e RF = 9; ( 4 ─ 1 ) mod 3 ≡ 0 and RF mod 7 ≡ 2;
Then the remainder of the division of N by 7 is 2.

 Additional example:

N = 124,934,652; ( ─ 52 mod 7 + 4 ) mod 7 ≡ 1; → 1249316;( ─ 16 mod 7 + 9 ) mod 7 ≡ 0; → 12403; ( ─ 3 mod 7 + 2 ) mod 7 ≡ 6; → 164;( ─ 64 mod 7 + 0 ) mod 7 ≡ 6; → 61; 7Ɨ61 and 7ƗN

The remainder: ( n ─ 1 ) mod 3 ≡ 1; 2 . 61 mod 7 ≡ 3; the remainder of the division of N by 7 is 3.