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# उपसूत्र ५. वेष्टनम्

## (Upasūtra 5. veṣṭanaṃ) - By osculation.

The Upasūtra: veṣṭanaṃ (By osculation) is used for divisibilty checks, for any number. It involves a technique termed 'osculation', with the 'veṣṭanah' (osculator) of the Divisor.

Note that, divisibility checks for numbers ending with 2 and 5 are obvious, and all even numbers (numbers ending with 2, 4, 6, 8, 0) are easily derived. So, the problem of divisibility-check arises for numbers ending with (1, 3, 7 and 9). This is where the Upasūtra is implemented. As such, it is an added technique (along with the conventional techniques) for divisibility checks, and completes the pattern for a number ending with any digit.

This Upasūtra adopts two different techniques for two different situations. Using positive osculators, through positive osculation - for numbers ending with 9 (and thus, for numbers ending with 1, 3 and 7); and using negative osculators, through negative osculation - for numbers ending with 1 (and thus, for numbers ending with 3, 7 and 9).

Divisibility check using positive osculation
As an illustration, let us use this Upasūtra for checking divisibility of:
304 by 38
 Steps 304 by 38 1. Exhaust the conventional techniques for 2, 5 and 10, to arrive at the case that needs osculation. In this case, 304 and 38 are both even numbers and divisible by 2. So, we divide both numbers by 2 to arrive at the need of osculation, to get: 152 by 19 Note that, 304 will be divisible by 38, if 152 is divisible by 19 2. Find the positive osculator of the Divisor (or new Divisor). In this case, the new Divisor is 19 And, the Osculator for 19 is: 2 (Calculating a Osculator is explained below) 3. Start the osculation from the right-most digit (Unit's place) of the Dividend. Multiply it with osculator. In this case, the first digit is 2 And, 2 × 2 = 4 4. Subtract as many Divisors from the new number as possible, and add with the next digit. In this case, 4 is less than 19, so there is no subtraction. Adding with next digit, we get: 4 + 5 = 9 5. Repeat steps 3 and 4, till all the digits are exhausted. In this case, 9 × 2 = 18. No subtraction required.  For the next digit, 18 + 1 = 19. Subtracting 19, we get 0  All digits are exhausted. 6. If the last derived number is 0, or a multiple of the Divisor - then the Dividend is divisible by the Dividend. In this case, the last derived number is 0 So, 304 is divisible by 38  On reaching the last derived number, it is not required to subtract the Dividends anymore - if we get the clue that it is a multiple of the Dividend or not.
Obviously, we need to understand Positive Osculator before moving ahead:
 For a number ending with 9, the Positive Osculator is: 1. Drop the last digit of 9. 2. For the remaining number, add 1 with it.  Now, if the Divisor does not end with 9 then: For a number ending with 3: Multiply with 3, and then find the osculator; For a number ending with 7: Multiply with 7, and then find the osculator; For a number ending with 1: Multiply with 9, and then find the osculator.  As examples, Osculator of 3 is: 1 (3 × 3 = 9, and 0 + 1 = 1) Osculator of 7 is: 5 (7 × 7 = 49, and 4 + 1 = 5) Osculator of 11 is: 10 (11 × 9 = 99, and 9 + 1 = 10) Osculator of 29 is: 3 (2 + 1 = 3) Osculator of 33 is: 10 (33 × 3 = 99, and 9 + 1 = 10) Osculator of 119 is: 12 (11 + 1 = 12)  Note: That the Osculator of two numbers can be the same (like 11 and 33 above).
So, for a practitioner of Vedic Mathematics, for divisibility of:
2049 by 29
 The Osculator is: 2 + 1 = 3  After the first digit: 2   0   4   9         2 (Note, 9 × 3 = 27, and 27 + 4 = 31. But 31 is greater than 29, so we subtract to get 2)  After the second digit: 2   0   4   9     6   2 (Note, 2 × 3 = 6, and 6 + 0 = 6)  After the third digit: 2   0   4   9 20  6   2 (Note, 6 × 3 = 18, and 18 + 2 = 20)  20 is not 0, or multiple of 29 Thus, 2049 is not divisible by 29
Again, for divisibility of:
23310 by 370
 Clearly, both Dividend and Divisor are multiple of 10 Dividing both by 10, we get 2331 and 37 As such, we need to check divisibility of 2331 by 37  The Osculator is: 26 (Note, 37 × 7 = 259, and 25 + 1 = 26)  After the first digit: 2   3   3   1         29 (Note, 1 × 26 = 26, and 26 + 3 = 29)  After the second digit: 2   3   3   1     17  29 (Note, 29 × 26 = 754, and 754 + 3 = 757. Clearly, 37×20 = 740, and subtracting it from 757, we get: 17)  After the third digit: 2   3   3   1 444 17  29 (Note, 17 × 26 = 442, and 442 + 2 = 444)  444 is a multiple of 37 (Note, 37 × 12 = 444) Thus, 23310 is divisible by 370

Divisibility check using negative osculation
As an illustration, let us use this Upasūtra for checking divisibility of:
168 by 21
 Steps 168 by 21 1. Exhaust the conventional techniques for 2, 5 and 10, to arrive at the case that needs osculation. In this case, conventional techniques do not apply. 2. Find the negative osculator of the Divisor (or new Divisor). In this case, the Divisor is 21 And, the Osculator for 21 is: 2 (Calculating a Osculator is explained below) 3. Arrange the Dividend in a manner that alternate digits from right are negative, and expressed as Rekhanks. In this case, it is arranged as 168 4. Start the osculation from the right-most digit (Unit's place) of the Dividend. Multiply it with osculator. In this case, the first digit is 8 And, 8 × 2 = 16 5. Subtract as many Divisors from the new number as possible, and add with the next digit. In this case, 16 is less than 21, so there is no subtraction. Adding with next digit, we get: 16 + 6 = 10 6. Repeat steps 3 and 4, till all the digits are exhausted. In this case, 10 × 2 = 20. No subtraction required.  For the next digit, 20 + 1 = 21. Subtracting 21, we get 0  All digits are exhausted. 7. If the last derived number is 0, or a multiple of the Divisor - then the Dividend is divisible by the Dividend. In this case, the last derived number is 0 So, 168 is divisible by 21  On reaching the last derived number, it is not required to subtract the Dividends anymore - if we get the clue that it is a multiple of the Dividend or not.
The technique followed by this Upasūtra uses Rekhanks & Vinculum Numbers (discussed here »). And obviously, we need to understand Negative Osculator before moving ahead:
 For a number ending with 1, the Negative Osculator is: 1. Drop the last digit of 1. 2. The remaining number is the Osculator.  Now, if the Divisor does not end with 1 then: For a number ending with 3: Multiply with 7, and then find the osculator; For a number ending with 7: Multiply with 3, and then find the osculator; For a number ending with 9: Multiply with 9, and then find the osculator.  As examples, Osculator of 3 is: 2 (3 × 7 = 21, thus: 2) Osculator of 7 is: 2 (7 × 3 = 21, thus: 2) Osculator of 11 is: 1 Osculator of 29 is: 26 (29 × 9 = 261, thus: 26) Osculator of 33 is: 23 (33 × 7 = 231, thus: 23) Osculator of 119 is: 107 (119 × 9 = 1071, thus: 107)  Note: That the Osculator of two numbers can be the same (like 11 and 33 above).
So, for a practitioner of Vedic Mathematics, for divisibility of:
6724 by 82
 Clearly, both Dividend and Divisor are multiple of 2 Dividing both by 2, we get 3362 and 41 As such, we need to check divisibility of 3362 by 41  The Osculator is: 4 Arranging the Dividend, we get: 3362  After the first digit: 3   3   6   2         2 (Note, 2 × 4 = 8, and 8 + 6 = 2)  After the second digit: 3   3   6   2     11  2 (Note, 2 × 4 = 8, and 8 + 3 = 11)  After the third digit: 3   3   6   2 41  11  2 (Note, 11 × 4 = 44, and 44 + 3 = 41)  41 is obviously a multiple of 41 Thus, 6724 is divisible by 82
Again, taking a previous example, for divisibility of:
23310 by 370
 Clearly, both Dividend and Divisor are multiple of 10 Dividing both by 10, we get 2331 and 37 As such, we need to check divisibility of 2331 by 37  The Osculator is: 11 (Note, 37 × 3 = 111, thus: 11) Arranging the Dividend, we get: 2331  After the first digit: 2   3   3   1         8 (Note, 1 × 11 = 11, and 11 + 3 = 8)  After the second digit: 2   3   3   1     17  8 (Note, 8 × 11 = 88, and 88 + 3 = 91. Clearly, 37×2 = 74, and subtracting it from 91, we get: 17)  After the third digit: 2   3   3   1 185 17  8 (Note, 17 × 11 = 187, and 187 + 2 = 185)  185 is a multiple of 37 (Note, 37 × 5 = 185) Thus, 23310 is divisible by 370
But, why does it work? For this Upasūtra (veṣṭanaṃ), let us consider the following:
 For both the positive and negative osculation, the technique implemented is a variation of parāvartya yojayet - using a 'Transposition', which is the Osculator.  This Upasūtra obviously works because the entire process is a method of division - where the Quotient is being ignored.

Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »

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