# DEVELOPING DIVISIBILITY RULES FOR ANY DIVISOR

Yesterday, I read a post here in Facebook about a 12-year old Nigerian boy, Chika Ofili, who received a special recognition for discovering a new formula for divisibility by 7, while doing his holiday assignment. The book they are using listed divisibility rules for 2,3,4,5,6 8 and 9 “but had no easy or memorable test listed for checking divisibility by 7.”

The boy really must be congratulated because he independently devised his method and did not merely rely on the conventional way Mathematics is taught – giving learners formulas to memorize.

While some may doubt Chika’s accomplishment given his young age, I believe he is really capable of that discovery.

I remember my own experience when I was a bit older than Chika, in 1965 or 1966 when the Indian Math wizard Shakuntala Devi visited the Philippines. Most us in the second batch of Philippines Science High School scholars (Pisay DOS) became instant Math lovers, inspired by Devi’s awesome calculating prowess. Devi mentioned about perfect numbers – those having the sum of their proper factor being equal to itself, like 6 ( 1+ 2 + 3 = 6) and 28 ( 1 + 2 + 4+ 7 + 14), I literally dreamed about perfect numbers and after several days, came up with a formula enabling me to find the next two perfect numbers 496 and 8128. Did I discover the formula for perfect numbers? There was no way for me to find out then.

In 2006, after 40 years and with the internet, I found out that my formula was correct but there were already 44 known perfect numbers by that time (now we have 51) and those 4 that I knew were discovered as early as the 4th century B.C. by Euclid.

Going back to Chika’s “discovery”, I felt a little sad that Chika and his teachers have not heard of Vedic Mathematics (VM). Probably because of its Oriental origin, its acceptance into the Western educational system is very slow. The divisibility method as described by Chika, was already published as early 1965 in the book Vedic Mathematics by Sri Bharati Tirthaji and later in Kenneth Williams’ Vedic mathematics Teacher’s manuals. As claimed by Tirthaji, VM came from the ancient Sankrit writing, the Vedas, and hence several hundred years old..

According to Tirthaji, 5 is the Ekadhika (one more than the one before) for 7 because 7 x 7 = 49. He wrote, “suppose we do not know and have to determine whether 21 is divisible by 7. We multiply the last digit, i.e. 1 by the Ekadhika (or Positive Osculator, i.e. 5) and add the product, i.e. 5 to the previous digit, i.e. 2 and thus get 7. This process is technically called Osculation. And if, the result of osculation is the divisor itself (or a repetition of a previous result), we say that the given original dividend 21 is divisible by 7.”

Compare this method with the procedure commonly taught in Philippine schools today: remove the last digit of the divisor, double it and subtract the doubled number from the remaining digits. If the result is a number divisible by 7 ( -7, 0, 7, 14, etc) then original dividend is divisible by 7. If we start with 21, when we remove the last digit 1, double it to get 2 and then subtract 2 from the remaining digit 2, we will get 0.

Do you find any similarity between the two methods?
The second rule is also Osculation with 2 as the negative osculator of 7.

Osculation is an ingenious Vedic Math technique to quickly determine if a number is evenly divisible by any divisor. This is in sharp contrast with the conventional system where there is a separate rule for every divisor.

To understand Osculation we need to consider two facts:

1) Adding or removing the ending zeroes in a number does not alter its divisibility by a divisor. Examples, 7, 70 and 700 are all divisible by by 7.

2) Adding or subtracting any multiple of the divisor does not affect the divisibility of a number. Examples: since 140 is divisible by 7, 140 + 7 or 147 is also divisible by 7 and 140 -28 = 112 is also divisible by 7.

Osculation is a “short cut” of the general strategy to “destroy” or make the last digit zero. And then ignore or remove it. It is very easy to make the last digit zero by using 21 which is 3 x 7. If the last digit is 1, we just subtract 21. If the last digit is 2 we will deduct 2 x 21 = 42, etc.

Let us try with some examples:

91: 91 – 21 = 70; divisible

102: 102 – 2 x 21 = 102 – 42 = 60; not divisible

133: 133 – 3 x 21 = 133 – 63 = 70; divisible

1001: 1001 – 21 = 980; 980 – 80 x 21 = 980 -1680 = – 700; divisible

3472: 3472 – 2 x 21 = 3472 – 42 = 3432; 3430 – 30 x 21 = 3430 – 630 = 2800; divisible

Using osculation we have:

91: 9 – 2 x 1= 9 -2 = 7; divisible

102: 10 – 2 x 2= 10 – 4 = 6; not divisible

133: 13 – 2 x 3 = 13 – 6 = 7; divisible

1001: 100 – 2 x 1 = 98; 9 – 2 x 8 = 9 – 16 = -7; divisible

3472: 347 – 2 x 2 = 347 – 4 = 343; 34 – 2 x 3 = 34 – 6 = 28; divisible.

Having seen how deducting multiples of 21 can eliminate the last digits successively we can now discuss how 5 is used. As we mentioned earlier, 5, the Ekadhika of 7, comes from 7 x 7 = 49. 49 can be written as 50 – 1.

Now using the examples we used above, we will illustrate how we can use (50 – 1) eliminate the last digits until we arrive at number clearly divisible by 7:

91: 91 + 1 x (50 – 1) = 91 + (50 – 1) = 140; divisible

102: 102 + 2 x (50 – 1) = 102 + (100 – 2) = 200; not divisible

133: 133 + 3 x (50 – 1) = 133 + (150 – 3) = 280; divisible

1001: 1001 + 1 x (50-1) = 1001 + (50 – 1) = 1050; 1050 + 50(50 – 1)= 3500; divisible

3472: 3472 + 2 x (50 – 1) = 3472 + (100 – 2) = 3570; 3570 + 70x (50 – 1)= 7000; divisible

Now let’s apply the Osculation method which Chika discovered independently:

91: 9 + 1 x 5 = 14; divisible

102: 10 + 2 x 5 = 20; not divisible

133: 13 + 3 x 5 = 13 + 15 = 28; divisible

1001: 100 + 1 x 5 = 105; 10 + 5 x 5 = 35; divisible

3472: 347 + 2 x 5 = 357; 35 + 7 x 5 = 70; divisible.

For 31 and 41, we can use 3 and 4 respectively as negative osculator; for 19, 29 and 39, we can use 2, 3 and 4 as positive osculators.

We can multiply number like 13 and 23 by 3 to get 39 and 69 and use 4 and 7 respectively as positive osculators, We can also multiply them by 7 to get 91 and 161 and use 9 and 16 as negative osculators.

The same procedure can be used for divisors ending in 7 like 17, 37, etc.

A more extensive discussion about Osculation can be found in my forthcoming book, “30 Master Strategies in Computing”.