A FALSE DIVISIBILITY RULE

THE EXPERTS’ PREFERRED PROCEDURE

Among the existing procedures to verify divisibility by 7 of any integer the one most cited by experts may be described this way:

 “To determine if a number is divisible by 7, take the last digit off the number, double it and subtract the doubled number from the remaining number. If the result is evenly divisible by 7 (e.g. 14, 7, 0, -7, etc.), then the number is divisible by seven. This may need to be repeated several times.” http://www.aaamath.com/div66_x7.htm

According to the above-mentioned description, its application demands two calculations (one multiplication and one subtraction) in order to take off successively each final digit of the tested number. It is applied very slowly especially to large numbers. For example, a ten-digit number requires sixteen calculations.

Sometimes this procedure is referred to as a rule, sometimes as a trick. It is not a rule of divisibility in the strict sense and much less a trick.

It is not a rule in the strict sense because the definition of “divisibility rule” includes the adjectives “shorthand”, “shortcut” or the equivalent. Although it is a mathematical rule because it is efficacious, it is not a rule of divisibility by definition because it is not efficient.

It is not a trick because Mathematics is not magic. Some experts and tutors present this procedure as a trick and, in order to keep the mystery, they do not explain why it works; and the “why it works” of this procedure is extremely primary.

To illustrate what I said let me show the practice of that procedure and a better alternative:

 

                                                     N = 696,816                                       696,816

                                                                 ─ 126                                              ─ 56          

                                                            696,690                                       696,760

                                             ─ 1,890                                           ─ 560

                                                          694,800                                       696,200

                                                         ─ 16,800                                        ─ 4,200

                                                          678,000                                       692,000

                                                       ─ 168,000                                     ─ 42,000

                                                          510,000                                      650,000

 

Excluding the zero digits, that took the places of the final digits removed in the process, the results are 51 and 65 that are not multiples of 7 and are equivalent in modulo 7; 7Ɨ51 and 7Ɨ 65 then 7ƗN.

I showed two alternatives (none of them is, by definition, a rule of divisibility by 7) side by side to put in evidence some facts:

1) In a two-digit multiple of 7 the tens are always the double of the ones.

Examples: 6 . 2 = 12 → 126 and 6 . 2 = 12; 12 mod 7 ≡ 5 → 56; 126 mod 7 ≡ 56 mod 7

                     9 . 2 = 18 → 189 and 9 . 2 = 18; 18 mod 7 ≡ 4 → 49; 189 mod 7 ≡ 49 mod 7

Why to do the multiplication 6 . 2, for example, if the resulting 12 mod 7 is equivalent to 5 mod 7 that may be deduced directly and quickly by the elementary 7 times table?

2) Although it puts in evidence the primarity of the procedure, it is simpler and quicker to use the second alternative: it is based directly on the 7 times table.

3) I included the zeros to demonstrate that, opposing some experts’ texts, the process preserves the value mod 7 of the original number in each step of the process.

4) It is easier and quicker to figure out the digit of the tens of a two-digit number multiple of 7 instead of multiplying one digit by 2 before each subtraction. The second alternative involves only subtractions of one-digit numbers easily deduced.

In sum this rule whose application is extremely slow consists of successive subtractions of multiples of 7; very rudimental indeed. It does not deserve the huge amount of papers written by such a great number of experts. None of these papers ever mentioned the successive subtractions of multiples of 7! Why not?

I think it is embarrassing that a few professional mathematicians try to impose and defend mathematical rules that are not, by definition, rules of divisibility by 7 just because they do not want to accept the real rules created by me.

My next post will be about another “rule” mentioned by some experts.