OPPOSITION TO MY CLAIM

OTHER FALSE RULES

 

A professional mathematician sent me some web addresses to contest the claim that I created the first rule of divisibility by 7 of the History of Number Theory. Now it is time to analyze the non-rules of divisibility by 7 mentioned in those web addresses and some others that were not mentioned.

I repeat: I have been trying to create a real rule of divisibility by 7 since the beginning of the 1990’s and I saw a great variety of procedures that are not real rules of divisibility by 7 according to the definition of “divisibility rule” mentioned by Wikipedia:

http://en.wikipedia.org/wiki/Divisibility_rule

My search was not continuous. It was intermittent and sometimes I abandoned it. However, after approximately 15 years of search, in 2005, I created my first method that I made public through a web site that is inactive now. In the same year, naming it as “Moura Velho Method” I created the first rule of divisibility by 7 that fits into the definition of divisibility rule.

Studying to demonstrate the “Why it works” of my first rule I learned some details of Pascal’s theorem 2.5 that he demonstrated in 1654 and that introduces a criterion for divisibility by 7. Pascal tried intensely to create a real rule of divisibility by 7, but he failed. His criterion for divisibility by 7 applied to a six-digit number demands the multiplication, from left to right, of each digit of N respectively by 5,4,6,2,3 and 1; if the sum of the products obtained results in a number that is a multiple of 7 then the tested number is also a multiple of 7.

 

 

N = 389,613; SP = 5 . 3 + 4 . 8 + 6 . 9 + 2 . 6 + 3 . 1 + 1 . 3 = 15 + 32 + 54 + 12 + 3 + 3 = 119

7|119 and 7 |N; Pascal would repeat the application of his criterion to confirm that 7|N:

2 . 1 + 3 . 1 + 1 . 9 = 14.

Pascal’s criterion for divisibility by 7 does not fit into the definition of “divisibility rule” because its application is very slow. References to Pascal’s great efforts to create a rule of divisibility by 7 may be found in the “HISTORY OF BINARY AND OTHER NONDECIMAL NUMERATION” written by Prof. Anton Glaser that may be accessed through this web address:

http://www.eipiphiny.org/books/history-of-binary.pdf

Many experts mention the procedure created by Pascal but they rarely mention his name. One of the addresses sent to me by the professional mathematician shows the application of Pascal’s criterion for divisibility by 7 without mentioning the name of the famous French mathematician.

Pascal’s criterion is cited in one of the addresses sent to me by the professional mathematician initially mentioned. That is the reason why I started my analyses by Pascal’s criterion that certainly does not fit into the definition of divisibility rule.

The experts’ preferred procedure to test divisibility by 7 was analyzed in my previous post (a false rule) and it also does not fit into the definition of divisibility rule.

 

N = 389,613; 389,613 ─ 63 = 389,550; 389,550 ─ 1050 = 388,500; 388,500 ─ 10500 = 378,000;

378,000 ─ 168,000 = 210,000; 7|210,000 and 7|N

There is one method of divisibility by 7 that was created by Prof. Gustavo Toja. The author do not claim that his method is a rule of divisibility. It certainly is a mathematical rule used to test divisibility by 7 but it also does not fit into the definition of divisibility rule. Linked to the same method the names of Martin Gardner and Alexander Bogomolny were cited.

N = 389,613; 38   96   13 → 3   2   6 → 6    23; 6   5 → 56; 7|56 and 7|N

3 2 6 from (38 ─ 3) (98 ─ 96) and (13 ─ 7) → inversion to 6 23

6 5 from (6 ─ 0) and (28 ─ 23) → inversion to 56

It reaches the correct result but I consider it complicated. See it in details accessing:

 http://www.sciencenews.org/view/generic/id/6193/description/Divisibility_by_Seven

My best rule of divisibility by 7 (see my post) also works with pairs of digits and fits into the definition of divisibility rule. Compared to Toja’s method it is applied in a much simpler way and it is incomparably quicker. Conclusion: Toja’s method cannot be used to object my claim.

 

N = 389,613; ( ─ 13 mod 7 + 9) ≡ 3 → 3836; ( ─ 36 mod 7 + 3) mod 7 ≡ 2 → 28; 7|28 and 7|N

It is very easy to apply my best rule entirely through mental calculation.

Prof. Gustavo Toja is Brazilian like me but he has a degree in Mathematics: this explains why his work had the receptiveness it had among the experts.

Now is the time to address other procedures used to test the divisibility of a number by 7 that were not mentioned to contest my claim.

There is one procedure that performs the algebraic sum of the numbers formed by the blocks of three digits of a number alternating the signs plus and minus applied to each block. If the result is a multiple of 7 then the tested number is a multiple of 7.

N = 389,613; 613 ─ 389 = 224; sometimes the result is a three-digit number and the solution needs an additional work. If N is larger, the application of the procedure is even more time-consuming.

This procedure cannot be considered a rule of divisibility by 7 because its application is very slow and it does not fit into the definition of divisibility rule.

Better than this procedure is my simplest rule (see the post). It is simpler and quicker than the procedure under analysis with one advantage: if 7ƗN, its result in modulo 7 is equivalent to the remainder of N divided by 7.

N = 389,613; ( ─ 38 mod 7 + 61 ) mod 7 ≡ 2 the tens

( ─ 9 mod 7 + 3) mod 7 ≡ 1 the ones → 21; 7|21 and 7|N

There is a procedure that consists of successively multiplying by 3 the first digit of N and adding the result to the next digit. The first digit is eliminated and the new first digit is submitted to the same procedure until the penultimate digit is reached. If the last two-digit number is a multiple of 7 then N is also a multiple of 7. Otherwise, the last two-digit number in modulo 7 is the remainder of N divided by 7.

N = 389,613; ( 3 . 3 + 8 ) mod 7 ≡ 3 → 39,613; ( 3 . 3 + 9 ) mod 7 ≡ 4 → 4,613;

( 4 . 3 + 6 ) mod 7 ≡ 4 → 413; ( 4 . 3 + 1 ) mod 7 ≡ 6 → 63; 7|63 and 7|N

This procedure is not a rule of divisibility by the same reasons appointed regarding the previous procedure.

Better and quicker than this procedure is to deduce the digit that forms a two-digit number multiple of 7 with the first digit and get a new first digit by adding the deduced digit to the first and second digit in modulo 7. The procedure must be repeated until the penultimate digit of N is reached. The last result mod 7 determines if 7|N or the remainder if 7ƗN.

N = 389,613; ( 6 + 3 + 8) mod 7 ≡ 3 → 39,613; ( 6 + 3 + 9 ) mod 7 ≡ 4 → 4,613;

( 1 + 4 + 6) mod 7 ≡ 4 → 413; ( 1 + 4 + 1) mod 7 ≡ 6 → 63; 7|63 and 7|N

There is a procedure that consists of multiplying the first digit of N by 2 and adding the result to the number formed by the next two digits. The first digit is eliminated and the new first digit must be submitted to the same procedure until the antepenultimate digit is submitted to the procedure. Its application is very slow and so it cannot be considered a rule of divisibility according to the definition of divisibility rule.

N = 389,613; 2 . 3 + 89 = 95 → 95,613; 2 . 9 + 56 = 74 → 7,413; 2 . 7 + 41 = 55 → 553;

2 . 5 + 53 = 63; 7|63 and 7|N

The use of Modular Arithmetic turns quicker the application of the procedure:

( 2 . 3 + 89 ) mod 7 ≡ 4 → 4,613; ( 2 . 4 + 61 ) mod 7 ≡ 6 → 63; 7|63 and 7|N

Perhaps this version, created by me, turns the procedure into a rule of divisibility because its applications is very quick. It may be performed entirely through mental calculation.

Why did not anybody thought of this before?

This procedure is superior to the experts’ preferred procedure to test divisibility by 7 because its result in modulo 7 is equivalent to the remainder of N divided by 7 if 7ƗN.

See my post “Applying Pascal’s multipliers” to know a better and quicker variation of this procedure.

The last procedure I will comment consists of multiplying the last digit of N by 5 and adding it to the two-digit number formed by the two following digits to the left. It must be repeated until the two leftmost digits are reached. Its application is very slow and cannot be considered a rule of divisibility according to the Wikipedia definition.

N = 389,613; 38961 + 15 = 38976; 3,897 + 30 = 3,927; 392 + 35 = 427; 42 + 35 = 77; 7|77 and 7|N

                              ( 3 . 5)                           ( 6 . 5)                  ( 7 . 5 )            ( 7 . 5)

Better than applying this procedure is to perform this way: Deduce the ones of a two-digit number multiple of 7 that is formed with the rightmost digit of N. Add the deduced digit to the two rightmost digits of N and represent the sum in modulo 7; the result is the new rightmost digit. Repeat the procedure until N is reduced to a two-digit number. If this last number is a multiple of 7 then N is also a multiple of 7.

389,613(5); ( 1 + 3 + 5 ) mod 7 ≡ 2 → 38962(1); ( 6 + 2 + 1 ) mod 7 ≡ 2 → 3892(1);

( 9 + 2 + 1 ) mod 7 ≡ 5 → 385(6); ( 8 + 5 + 6 ) mod 7 ≡ 5 → 35; 7|35 and 7|N

I do not remember of any other procedure used to determine if a number is divisible by 7. If any person wants to contest my claim presenting any procedure that was not examined in this post, please, let me know. Especially if the procedure may fit into the definition of divisibility rule.