VEDIC MATHS-65

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VEDIC MATHS

                           By OMKAR TENDOLKAR

Hello friends,

                      This is  post number 65 from the series of "Vedic maths" blogs. Here in this blog we will learn about "The Osculation Method for checking divisibility of number by Negative Osculator"

Concept of Osculator :

The concept of ‘Osculator’ is useful to check the divisibility of a given number by divisors ending with 9 or 1 or a multiple there of.

An osculator is a number defined for any number ending in 9 or 1 and is obtained from the number by a simple mechanism described in this chapter.

The use of osculators would be severely limited if only these two categories of numbers viz. ending with 9 or 1 were considered. Interestingly, numbers ending with 3 and 7 can also converted to numbers ending with 1 or 9 by a suitable multiplication. The same techniques can be used for all such numbers too.

Osculators are categorized into two main types viz. positive and negative depending on whether the number ends with 9 or with 1.

Reference: 

We had already learn about "Concept of Positive Osculator" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-62"

We had already learn about "The Osculation Method for checking divisibility of number by Positive Osculator" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-63"

We had already learn about "Concept of Negative Osculator" in our previous blog. If you have missed my last blog then please visit "VEDIC MATHS-64"

Negative Osculators :

The Osculator of a number ending in 1 is considered negative. It is obtained by 
  • Dropping 1
  • Remaining number is Negative Osculator
The Negative Osculator for 11 is 2, for 21 it is 2 and for 51, it is 5.

If the given number does not end in 1 but in 9 or 3 or 7, we can multiply it by 9, 7 or 3 respectively, to convert it to a number which ends in 9 and then compute the Negative Osculator by dropping the 1.

Examples are given below of how to compute the Negative Osculators for various numbers.

Once the Negative Osculator of a number has been computed, it is a simple matter to check the divisibility of any given number by it.


The Osculation Method for Negative Osculator :

This method will help us find the divisibility rule for any number.

Examples :

1. Check whether 6603 divisible by 31 or not.

Again we find the Negative Osculator of 31.

  • Dropping 1
  • Remaining number is Negative Osculator i.e. 3

Negative Osculator of 31 is 3.

Now we osculate 6603 with 3.

To osculate a number, we multiply its last figure by the Negative Osculator and then subtract the result from its previous figure.

So, say we have 6603.

660 - (3 × 3) 

= 651

Now we osculate again. We multiply 1 by 3 and subtract from 63.

= 65 - (1 × 3) 

= 62

Now we osculate again. We multiply 2 by 3 and subtract from 6.

6 - (2 × 3) 

= 6 - 6 

= 0

For a number to be divisible, the result of the osculation should be the divisor, zero or a repetition of a previous result. This 0 indicates that the number 6603 is divisible by 31.

Answer :
6603 is divisible by 31.

2. Check whether 11234 divisible by 41 or not.

Again we find the Negative  Osculator of 41.

  • Dropping 1.
  • Remaining number is Negative Osculator i.e. 4

Negative Osculator of 41 is 4.

We now osculate 11234 with 5.

To osculate a number, we multiply its last figure by the Negative Osculator and then subtract the result from its previous figure.

So, say we have 11234.

1123 - (4 × 4) 

= 1107

Now we osculate again. We multiply 7 by 4 and subtract from 110.

110 - (7 × 4)

= 82

Now we osculate again. We multiply 2 by 4 and subtract from 8.

8 - (2 × 4) 

= 0

This 0 indicates that the number 11234 is divisible by 41 completely.

Answer :
11234 is divisible by 41.

3. Check whether 2275 is divisible by 7 or not.

We first find the Negative Osculator of 7.

  • Here, given number 7 end with 7, we can multiply it by 3, to convert it to a number which ends in 1 i.e. 21.
  • Dropping 1
  • Remaining digit is Negative Osculator i.e  2

Negative Osculator of 7 is 2.

We now osculate 2275 with 2 

To osculate a number, we multiply its last figure by the Negative Osculator and then subtract the result from its previous figure.

Now we start osculating. So we have:

227 - (5 × 2) 

=  217

Now we osculate again. We multiply 7 by 2 and subtract from 21.

21 - (7 × 2) 

= 7

Since we got 7 (which is also the divisor) as the result of the osculation process, we can safely say that 2275 is divisible by 7. 

Answer :
2275 is divisible by 7.

4.  Check whether 464411 by 71. or not.

We first find the Negative Osculator of 71

  • Dropping 1
  • Adding 1 to remaining digit i.e. 6 gives 7.

Negative Osculator of 71 is 7.

We now osculate 464411 with 7.

To osculate a number, we multiply its last figure by the Negative Osculator and then subtract the result from its previous figure.

Now we start osculating. So we have:

46441 - (1 × 7) 

= 46434

Now we osculate again. We multiply 4 by 7 and subtract from 4643.

4643 - (4 × 7) 

= 4615

Now we osculate again. We multiply 5 by 7 and subtract from 461.

461 - (5 × 7) 

= 426

Now we osculate again. We multiply 6 by 7 and subtract from 42.

42 - (6 × 7) 

= 0

This zero indicates that the number 464411 is divisible by 71 completely.

Answer :
464411 is divisible by 71.


Solve the following example by Positive Osculation.

1. Check whether 315 is divisible by 21 or not.
Ans : 315 is divisible by 21.

2. Check whether 52793 is divisible by 31 or not.
Ans : 52793 is divisible by 31.

3.Check whether 945 is divisible by 21 or not.
Ans : 945 is divisible by 29.

4. Check whether 11234 is divisible by 41 or not.
Ans : 11234 is divisible by 41.

5. Check whether 210661 is divisible by 11 or not.
Ans : 210661is divisible by 11.

The simplicity of this method can be vouched from examples given above.

You may try following examples for checking divisibility by Positive Osculation Method.
       
1. Check whether 179445 is divisible by 61 or not.
2. Check whether 113465 is divisible by 11 or not.
3. Check whether 158661 is divisible by 51 or not.
4. Check whether 179456 is divisible by 41 or not.
5. Check whether 11669 is divisible by 31 or not.

In next blog we will discuss about "2-digits Addition from left to right".     

Are you excited for this?...
Then, please wait for it.
I will post my new blog in next week.

We will meet very soon through our next  blog. Till that stay connected, stay healthy and stay safe.

Thanks

for giving your valuable time.

Good day😊

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