OTHER QUICK RULES

OTHER QUICK RULES

I will present these rules just to show how my research provided various possibilities to the creation of numerous quick rules for divisibility by 7.

Practice turns quick and easy the application of my rules. At first, as any new knowledge, it may look somehow complicated. Knowing the multiplication table of seven and having some skill with mental calculations are the requirements to apply my rules.

Remember that, without success, researchers have been trying to create a real rule of divisibility by 7 since the beginning of the first millennium (Talmud) according to the History Of Number Theory. When I say real rule I refer to a rule that fits into the definition of divisibility rule according to Wikipedia:

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

I think that I presented my best rules in my previous posts and that the explanations of why they work are enough to understand the “why they work” of the next rules. Therefore, I will present only the respective algorithm and some numerical examples. If necessary, I will highlight important details.

Rule 1

N = abc,def → abxc, def

─ ab mod 7 + x + de = de’→ de'f; de'xf; R = ─  de' mod 7 + x

 R = ─ de’ mod 7 + x; if 7|R then 7|N

N = 946,132 → 9456, 132 → 222 → 2242 → 10

─ 94 mod 7 + 5 + 13 = 22; R = ─ 22 mod 7 + 4 = 10; 7Ɨ10 and 7ƗN

Remainder: (R mod 7 . 4) mod 7; (10 . 4) mod 7 ≡ 5 = r (remainder of the division N/7)

The algorithm must be repeated until the last period is reached. It uses repetitively the multipliers 462 to each period of N. It is easy to extend its application to larger numbers.

This rule may be simplified even more:

N = abc,def 

( ─ ab mod 7 + x + de ) mod 7 = e' → e'f; if 7|e'f then 7|N 

N = 946,132 → 9456,132

( ─ 94 mod 7 + 5 + 13 ) mod 7 ≡ 1 → 12; 7Ɨ12 and r = 12 mod 7 ≡ 5 (remainder of N/7)

In this case the multipliers applied to the 5 initial digits are: 46231.

Rule 2

N = abc.def → aybc, dyef

─ bc mod 7 + a + y + ef; R = ─ ef’ mod 7 + d + y. if 7|R then 7|N

N = 946,132 → 9146, 1432 → 145 → 1445

─ 46 mod 7 + 9 + 1 + 32 = 45; ─ 45 mod 7 + 1 + 4 = 9; 7Ɨ9 and 7ƗN

Remainder: ─ R mod 7 ≡ r; ─ 9 mod 7 ≡ 5 = r (remainder of N/7)

The algorithm must be repeated until the last period is reached. It uses repetitively the multipliers 546 to each period of N. It is easy to extend its application to more extensive numbers.

Rule 3

N = a,bcd

(─ cd mod 7) . 3 + b → ab’; if 7|ab’ then 7|N

N = 946,132

[(─ 32 mod 7) . 3 + 1 ] mod 7 = b’ = 3 → 9463

[(─ 63 mod 7) . 3 +4 = b‘ = 4 → 94; 7Ɨ94 and 7ƗN

r = (94 mod 7 . 4) mod 7 ≡ 5 (remainder of N/7)

Regarding larger numbers the algorithm must be applied until the leftmost digit is reached.

The procedure to determine the remainder is the same of a previous rule, already presented, that works with pairs of digits.

Rule 4

N = a,bcd → a,bycd

─ (a + b + y) mod 7 + cd

N = 946,132; 9426132 → 6732 → 67032

─ (9 + 4 + 2) mod 7 + 61 = 67

─ ( 6 + 7 + 0) mod 7 + 32 = 33; 7Ɨ33 and 7ƗN

r = 33 mod 7 = 5 (remainder of the division N/7)

The algorithm must be applied repetitively to four digits each time until the last digit is reached.

The multipliers applied are: 6231. To reach this conclusion it is necessary some reasoning.

Rule 5

N = a,bcd → a,bxcd

[─ (x + c + d) mod 7 + a] mod 7

This algorithm must be applied from right to left. In each application, the last two digits must be eliminated.

N = 946,132; 61632

[─ ( 6 + 3 + 2 ) mod 7 + 6] mod 7 = 2 → 9421 → 94421

[─ (4 + 2 + 1 ) mod 7 + 9] mod 7 ≡ 2 → 24; 7Ɨ24 and 7ƗN

r = (24 mod 7 . 4) mod 7 ≡ 5 = r (remainder of N/7)

The procedure to determine the remainder is the same of a previous rule, already presented, that works with pairs of digits.

CONCLUSION

After I deciphered Pascal's criterion of divisibility by seven it became very easy to create quick rules of divisibility by seven. Of course, there are many other rules that I might present, but I think that the rules already presented are enough to illustrate my discoveries.

In my next post I will comment some feedbacks I received from professional experts.