Chika’s Rule for Divisibility by 7

“Multiply the last digit by 5 and add it to the remaining number.” This is the divisibility rule which was said to be “discovered” by a 12-year old Nigerian boy, Chika Ofili for which he was given recognition by his school.

Examples:

651: 65 + 1 x 5 = 70. 70 is divisible by 7, then 651 is divisible by 7.

1687: 168 + 7 x 5 = 203; 20 + 3 x 5 = 35. 35  is divisible by 7 then so are 203 and 1687.

1911: 191 + 1 x 5 = 196; 19 + 6 x 5 = 49. 49 is divisible by 7 then so are 196 and 1911.

For reasons which will be clear later, 196 and 1911 are also divisible by 49.

Note that for 49 we have 4 + 9 x 5 = 49.

2996: 299 +  6 x 5 = 329

          32 + 9 x 5 = 77

           7 + 7 x 5 = 42

           4 + 2 x 5 = 14

           1 + 4 x 5 = 21

           2 + 1 x 5 = 7

In this last example, we saw that as early as we got 77, we know that 2996 is divisible by 7. Continuing the process eventually ended in 7.

In my Nov 14, 2019  post about this rule, I wrote: , I felt a little sad that Chika and his teachers have not heard of Vedic Math(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 as 1965 in the book Vedic Mathematics by Sri Bharati Tirthaji and later in Kenneth Williams’ Vedic Mathematics Teacher’s manuals (intermediate and Advanced levels. As claimed by Tirthaji, VM came from the ancient Sankrit writing, the Vedas and hence, hundreds of 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 the Ekadhika of 7, 5 comes from 7 x 7 = 49. 49 is 50 – 1. Now using the examples we used above, we will 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.

Practice exercises:

Determine which of the following numbers are divisible by 7 using Chika’s rule.

1. 791
2. 946
3. 973
4. 1701
5. 1847
6. 7154
7. 13209
8. 17431
9. 42791
10. 54999

Answers to previous week’s exercises:

Identify which of the following numbers are divisible by 7, and by 21.

1. 413 → divisible by 7
2. 692 → not divisible
3. 1932 → divisible by 7 and 21
4. 2961 → divisible by 7 and 21
5. 8882 → not divisible
6. 9583 → divisible by 7
7. 16,387 → divisible by 7
8. 23,884 → divisible by 7
9. 49,161 → divisible by 7 and 21
10. 71, 645 → divisible by 7