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Divisibility Tests (Vedic Mathematics)

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1. Osculators and Osculation

In Vedic Mathematics, testing divisibility is very easy and there is a general rule for it. The concept of osculators or “vestanas” is used for testing divisibility, and they are calculated by using the Vedic Mathematics sutra, called the “ekadhika sutra”. The process by which one tests the divisibility is called as “Vestanas” or “osculation”.
Watch the following video to learn how to find out the osculators of numbers:
Solved Examples
Example 1:
Calculate the osculator of 189.
Solution:

Example 2:
Calculate the osculator of 143.
Solution:
143 × 3 = 429

Example 3:
Calculate the osculator of 87.
Solution:
87 × 7 = 609

Example 4:
Calculate the osculator of 91.
Solution:
91 × 9 = 819

These osculators act as the building blocks for the divisibility test process. They are used to osculate the number in order to reach the conclusion that whether the number is divisible or not.
In order to understand the process of osculation, go through the following video.
Solved Examples
Example 1:
Osculate 1982 with the osculator of 39.
Solution:

Osculated result = 198 + 2 × 4 = 198 + 8 = 206
Example 2:
Osculate 832 with the osculator of 43.


Solution:
43 × 3 = 129

Osculated result = 83 + 2 × 13 = 83 + 26 = 109
Summary:
  • Osculators of numbers ending with 9: Drop the last digit and add 1 to the remaining digits.
  • Osculators of numbers ending with 3: Multiply the number by 3. Drop the last digit 9 and add 1 to the remaining digits.
  • Osculators of numbers ending with 7: Multiply the number by 7. Drop the last digit 9 and add 1 to the remaining digits.
  • Osculators of numbers ending with 1: Multiply the number by 9. Drop the last digit 9 and add 1 to the remaining digits.
  • Osculating abcd with p, one gets (abc) + dp.
Note:
A number osculated by its osculator gives the same number or a multiple of it.
For e.g., 29 (osculated by 3) gives 9 × 3 + 2 = 27 + 2 = 29
Again, 33 (osculated by 10) gives 10 × 3 + 3 = 33

2. Divisibility Tests for Numbers Ending With 3 and 9

This divisibility test is based on the method of “Vestana”, which means osculation.
There is no conventional method to determine the divisibility by numbers ending with 9. However, Vedic Mathematics makes this possible.
Watch the following video to learn the method.
Solved Examples
Example 1:
Check whether 2774 is divisible by 19 or not.
Solution:

Step I:
Osculating 74 with osculator 2:
4 × 2 = 8
Osculated result = 8 + 7 = 15

Step II:
Osculating 15 with osculator 2:
5 × 2 = 10
Osculated result = 10 + 1 = 11
Adding digit to its left:
11 + 7 = 18

Step III:
Osculating 18 with osculator 2:
8 × 2 = 16
Osculated result = 16 + 1 = 17
Adding digit to its left:
17 + 2 = 19

Since 19 is divisible by 19, 2774 is divisible by 19.
The method to check the divisibility by numbers ending with 3 is not too different than what we learnt. To understand this method, watch the following video.
Note: In the above video, the first osculated result is written under the third digit from the right and not under the second digit from the right. This is done only to make the process less confusing.
Solved Example
Example 1:
Check whether 2795 is divisible by 43 or not.
Solution:

Step I:
Osculating 95 with osculator 13:
13 × 5 = 65
Osculated result = 65 + 9 = 74

Step II:
Osculating 74 with osculator 13:
13 × 4 = 52
Osculated result = 52 + 7 = 59
Adding digit above 74:
59 + 7 = 66

Step III:
Osculating 66 with osculator 13:
13 × 6 = 78
Osculated result = 78 + 6 = 84
Adding digit above 66:
84 + 2 = 86

Since 86 is divisible by 43, 2795 is divisible by 43.

3. Negative Osculators and Negative Osculation

Negative osculator is a form of osculator, which is also used to check divisibility of numbers. They are used to check the divisibility of those numbers, where the positive osculator comes out to be too high, especially, in the case of numbers ending with 7 and 1.
For example, the osculator of 67 is calculated as:
67 × 7 = 469
⇒ Osculator of 67 = 46 + 1 = 47
47 is too high to osculate any number with it.
Hence, the introduction of negative osculator was required for cases like these.
To understand the concept of negative osculators, go through the following video.
Solved Examples
Example 1:
Calculate the negative osculator of 81.
Solution:

Example 2:
Calcualte the negative osculator of 191.
Solution:

Example 3:
Calculate the negative osculator of 47.
Solution:

Example 4:
Calculate the negative osculator of 77.
Solution:

Note:
Positive osculator of a number + Negative osculator of a number = The number itself
This can be validated by calculating the positive and negative osculators of 17.
17 × 7 = 119
⇒ Positive osculator of 17 = 11 + 1 = 12
17 × 3 = 51
⇒ Negative osculator of 17 = 5
Positive osculator + Negative osculator = 12 + 5 = 17 = The number itself
Summary:

Number
Negative osculator
Positive osculator

Ending with 1
Drop the last digit 1. Remaining digits are the osculator.
Multiply by 9 and drop the last digit 9. Add 1 to the remaining digits.

Ending with 3
Multiply by 7 and drop the last digit 1. Remaining digits are the osculator.
Multiply by 3 and drop the last digit 9. Add 1 to the remaining digits.

Ending with 7
Multiply by 3 and drop the last digit 1. Remaining digits are the osculator.
Multiply by 7 and drop the last digit 9. Add 1 to the remaining digits.

Ending with 9
Multiply by 9 and drop the last digit 1. Remaining digits are the osculator.
Drop 9 and add 1 to the remaining digits.
Hence, it can be concluded that
  • Negative osculators of numbers ending with 1 and 7 < Positive osculators of numbers ending with 1 and 7
  • Positive osculators of numbers ending with 3 and 9 < Negative osculators of numbers ending with 3 and 9
Always remember to use:
  • Positive osculator process for numbers ending with 3 and 9
  • Negative osculator process for numbers ending with 1and 7
Now look at the following video to understand the osculation process for negative osculators:
Solved Examples
Example 1:
Osculate 147 by the negative osculator of 51.
Solution:

7 × 5 = 35
∴Osculated result = 14 − 35 = −21
Example 2:
Osculate 4791 by the negative osculator of 37.
Solution:

1 × 11 = 11
∴Osculated result = 479 − 11 = 468
Note:
Osculation with a negative osculator is also called negative osculation.

4. Divisibility Test for Numbers Ending With 1 and 7

One often finds it difficult to test the divisibility by numbers ending with 1 and 7. However, there is a method for this under Vedic Mathematics, which uses the concepts of negative osculators and osculating with them.
The following video details the process to check the divisibility by numbers ending with 1. Watch the video to understand the method.
Solved Example
Example 1:
Check whether 7776 is divisible by 81 or not.
Solution:

Osculating 7776 by the negative osculator of 81:
6 × 8 = 48
777 − 48 = 729
Osculating 729 by the negative osculator of 81:
9 × 8 = 72
72 − 72 = 0
Therefore, 7776 is divisible by 81.
The same method used for checking the divisibility by numbers ending with 1 is used for numbers ending with 7. The only difference is the manner in which the negative osculators are calculated.
Watch the following video to understand this method.
Solved Example
Example 1:
Check whether 3626 is divisible by 37 or not.
Solution:

Osculating 3626 by the negative osculator of 37:
6 × 11 = 66
362 − 66 = 296
Osculating 296 by the negative osculator of 37:
6 × 11 = 66
29 − 66 = −37
One can observe that −37 is divisible by 37.
Hence, 3626 is divisible by 37.
Special case to check divisibility by 7:
(i) Double the last digit of the number.
(ii) Subtract it from the number formed from the remaining digits.
(iii) Test whether the difference is divisible by 7 or not.
(iv) If the difference is a large number, then repeat steps (i) to (iii) on the difference again and again, till you get a simple number.
(v) If the difference is divisible by 7, then the number is also divisible by 7; otherwise, it not divisible by 7.
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