FOLLOW THESE STEPS

1 – Choose a sequence of 3 to 6 of Pascal’s
multipliers: 546231546…

2 – Arrange the best way to perform the
operations.

3 – Make use of the additive inverse mod 7 when
needed

4 – Create the algorithm

5 – Apply it.

If necessary look at my last post.

First Example:

1 - Chosen sequence: 231

2 – N = abc; 2 . a + bc

3 – In this case the additive inverse mod 7
will be used.

4 – The algorithm: - ( (x) + bc ) mod 7; if N
has more than one period the result of one period must be added to the next period
till the last period is reached.

5 – Application

N = 154; - ( (2) + 54 ) mod 7 ≡ Ø; 7|Ø → 7|N

N = 4.641; - 4 mod 7 ≡ 3; ( 3 + (5) +
41 ) mod 7 ≡ Ø; 7|Ø → 7|N

N = 823.424; - ( (2) + 23 ) mod 7 ≡ 3;
( 3 + (1) + 24 ) mod 7 ≡ Ø; 7|Ø → 7|N

N = 6.453.243; - 6 mod 7 ≡ 1; - ( 1 + (1) + 53 ) mod 7 ≡ 1; ( 1 + (4) +
43 ) mod 7 ≡ 6; 7ł6; 7łN*

*In this case 6 is the remainder of N divided by 7. Observe that I don’t
apply the additive inverse mod 7 to the last result. In this way, if N is not
divisible by 7, the last result will give us the remainder of the division of N by 7.

N = 38.453.952; - 38 mod 7 ≡ 4; - ( 4 +
(1) + 53 ) mod 7 ≡ 4; ( 4 + (4) + 52 ) mod 7 ≡ 4; 4 is the remainder of the
division of N by 7.

Second example:

1 – Chosen sequence: 1546

2 – N = a.bcd; (6 . cd + a) mod 7; N is
reduced to a’b

3 – Additive inverse mod 7 is not
necessary as a link.

4 – The algorithm: - (cd mod 7 + a ) mod 7 ≡ a’; it is applied from
right to left.

5 – Application:

N = 154 → Ø154; - (54 + Ø) mod 7 ≡ 2 = a’
→ a’b = 21; 7|21 and 7|N

N = 4.641; - ( 41 mod 7 + 4 ) mod 7 ≡ 5 =
a’ → a’b = 56; 7|56 and 7|N

N = 823.424; ( - 24 mod 7 + 3 ) mod 7 ≡ Ø
= a’ → a’b = Ø4; ( - 4 mod 7 + 8) mod 7 ≡ 4 = a’

→ a’b = 42; 7|42 and 7 |N

N = 6.453.243; ( - 43 mod 8 + 3) mod 7 ≡ 2
= a’ → a’b = 22; ( - 22 mod 7 + 4 ) mod 7 ≡ 3 = a’

→ a’b = 35; ( - 35 mod 7 + Ø ) mod 7 ≡ Ø =
a’ → a’b = Ø6; 7ł6 and 7łN*

*Coincidentally the result was the same of
the anterior rule but generally this rule does not result in the remainder.

N = 38.453.952; ( - 52 mod 7 + 3 ) mod 7 ≡
Ø = a’ → a’b = Ø9; ( - 9 mod 7 + 4 ) mod 7 ≡ 2 = a’

→ a’b = 25; ( - 25 mod 7 + 3 ) mod 7 ≡ 6 =
a’ → a’b = 68; 68 mod 7 ≡ 5; 7ł5 and 7łN

Changing what must be changed this rule
may be applied to divisibility by 11 and 13.

Third example:

1 – Chosen sequence: 462

2 – N = abc; 6 . a

_{1}b_{1}+ (x) + a_{2}b_{2 …}
3 – Additive inverse mod 7 is not
necessary as a link

4 – The algorithm: ( - a

_{1}b_{1}mod 7 + (x) ) mod 7 + a_{2}b_{2 ….}
5 – Application:

N = 154; ( - 15 mod 7 + (1) ) mod 7 ≡ Ø; 7|Ø and 7|N

N = 4.641; - [ ( (1) + 64 mod 7 ) + (2) ] mod 7 ≡ Ø; 7|Ø
and 7|N

N = 823.424; [ ( - 82 mod 7 + (6) ] mod 7 ≡ 1; - [ ( 1 + 42 ) mod 7 +
(1) ] mod 7 ≡ Ø; 7|Ø and 7|N

N = 6.453.243; - [ ( (5) + 45 ) mod 7 + (6) ] mod 7 ≡ 5; - [ ( 5 + 24 ) mod
7 ) + 6 ] mod 7 ≡ 5; 7ł5 and 7łN

N = 38.453.952; ( - 3 mod 7 + 2 ) mod 7 ≡ 6; [ -( 6 + 45 mod 7 ) + (6) ]
mod 7 ≡ 4; [ - ( 4 + 95 mod 7) + (4) ]
mod 7 ≡ 3; 7ł3 and 7łN

Changing what must be changed this rule may be applied to divisibility
by 11 and 13.

The remainder is the value of the digit that forms with the final result
a two-digit number multiple of 7.

Fourth example:

1 - Chosen sequence: 315

2 – N = abc; ab + c + (y)

3 – Additive inverse mod 7 is necessary

4 – The algorithm: - ( ab + c + (y) ) mod
7 …

5 – Application:

N = 154; - ( 15 + 4 + (2) ) mod 7 ≡ Ø; 7|Ø
and 7|N

N = 4.641; - ( 4 + (2) ) mod 7 ≡ 1; - ( 1
+ 64 + 1 + (4) ) mod 7 ≡ Ø; 7|Ø and 7|N

N = 823.424; - ( 82 + 3 + (5) ) mod 7 ≡ 4;
- ( 4 + 42 + 4 + (2) ) mod 7 ≡ 3; 7ł3 and 7łN

N = 6.453.243; - ( 6 + (3) ) mod 7 ≡ 5; -
( 5 + 45 + 3 + (5) ) mod 7 ≡ 5;

- ( 5 + 24 + 3 + (5) ) mod 7 ≡ 5; 7ł5 and
7łN

N = 38.453.952; - ( 3 + 8 + (4) ) mod 7 ≡
6; - ( 6 + 45 + 3 + (5) ) mod 7 ≡ 4; - ( 4 + 95 + 2 + (1) ) ≡ 3

Changing what must be changed this rule
may be applied to divisibility by 13.

The remainder is the value of the digit
that forms with the final result a two-digit number multiple of 7.

Fifth Example:

1 - Chosen sequence: 546231

2 – N = abcdef; 5 . a + 6 . bc + 2 . d +
ef

3 – Additive inverse mod 7 is not necessary

4 – The algorithm: ( - bc mod 7) + a + (y)
+ (x) + ef

5 – Application:

N = 154; - 54 mod 7 + 1 + 4 = 7; 7|7 and
7|N

N = 4.641; - 4 mod 7 + (5) + 41 = 49; 7|49
and 7|N

N = 823.424; - 23 mod 7 + 8 + (4) + (1) +
24 = 42; 7|42 and 7|N

N = 6.453.243; 6 + (- 53 mod 7) + 4 + (2)
+ (4) + 43 = 62; 7ł62 and 7łN;

remainder = 62 mod 7 ≡ 6

N = 38.453.952; 38 + (- 53 mod 7) + 4 +
(2) + (4) + 52 = 103; 7ł103 and 7łN

remainder = 103 mod 7 ≡ 5

This rule demands more attention and I
included it because it uses the whole sequence of Pascal’s multipliers.

I think that five examples are enough to
demonstrate that the creation of real rules for divisibility by 7 became very
simple and accurate using the knowledge presented in my anterior post. As the
creation of the rules follows the sequence of Pascal’s remainders ( I call them
multipliers) that was proved by himself I consider that additional proves are
not necessary. The examples presented confirm the validity of my reasoning.

There are other characteristics of
Pascal’s multipliers that may be explored to create other rules that work
perfectly well. That will be the matter for the next post.