THE CREATION OF THE FIRST "MOURA VELHO RULE"

 

HOW I CREATED MY FIRST RULE OF DIVISIBILITY BY 7

 

I started researching divisibility by 7 in the beginning of the 1990’s. It took me a long time to reach my best results. In 2005 I created my first method of divisibility by 7 and without noticing it I created the first rule of divisibility by 7 of the History of Number Theory that fits the definition of divisibility rule according to Wikipedia:

 

“… a shorthand way of determining whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits”.

 

 After my first rule, I created various other rules that are quick and accurate.

 

From now on, I will summarize the steps I followed to create my first rule. There were various lapses of time between the steps, some shorter some longer.

 

First step:

 

N = ab; insert “x” before “a” in a way that 7|xa → xab; S = x + a +b

 

If 7|N then 7|S; if 7ƗN then S mod 7 ≡ r (remainder of ab/7)

 

N = 42 → 142 → S = 1 + 4 + 2 = 7; N = 96 → 496 → S = 4 + 9 + 6 = 19; 19 mod 7 ≡ 5 ≡ r

 

Second step:

 

N = bc; insert “y” after “c” in a way that 7|cy → bcy; S = b + c + y

 

If 7|N then 7|S; if 7ƗN then S mod 7 ≡ 5r (5 times the remainder of bc/7)

 

N = 56 → 563 → S = 5 + 6 + 3 = 14; 52 → 521 → S = 5 + 2 + 1 = 8; 8  mod 7 ≡ 1 ≡ 5 . r mod 7

(1 . 3) mod 7 ≡ 3 = r (remainder of N/7)

 

Third step:

 

N = abc; insert “x” before “a” and “y” after “c” according to the previous steps:

 

xabcy; S = x + a + b + c + y

 

If 7|N then 7|S; if 7ƗN then S mod 7 ≡ 5r mod 7 (5 times the remainder of N/7)

 

Fourth step:

 

N = abc,def

 

After applying the three initial steps to each period add ─ S1  mod 7 to S2:

 

S1 = x + a + b + c + y and S2 = x + d + e + f + y

 

In the passage of one period to another, the previous sum must be converted to its additive inverse modulo 7 and added to the next sum till the rightmost period is reached.

 

─ S1 mod 7 + S2 (two periods)

 

─ ( ─ S1 mod 7 + S2 ) mod 7 + S3 (three periods)

 

The repetitive and cumulative application of the additive inverse mod 7 results in an alternation of the signs plus and minus regarding each sum performed.

 

Example:

 

N = 26,965,466

 

First period: 263; 2 + 6 + 3 = 11; ─ 11 mod 7 ≡ “3” (this result will be added to the digits of the second period)

 

Second period: ”3”  49656;  3 + 4 + 9 + 6 + 5 + 6 = 33; ─ 33 mod 7 ≡ “2” (this result will be added to the digits of the third period)

 

Third period: “2” 14663; 2 + 1 + 4 + 6 + 6 + 3 = 22; 7Ɨ22 and 7ƗN

(3 . 22) mod 7 ≡ 3 = r (remainder of N/7)

My next post will be about an algorithm that applies the same multipliers established by Pascal's 2.5 Theorem.