**Vedic Mathematics Newsletter No. 132**

A warm welcome to our new subscribers.

This issue’s article is by Kenneth Williams, and is titled “Osculation Magic”.

“We can also jump forwards or backwards through the osculators, getting every 2^{nd}, 3^{rd}, 4^{th} etc. one. And we can get specific osculators as required, or find the number of osculators in the cycle (which is also the number of digits in the recurring decimal for n/D). There are many other beautiful and astounding properties within these osculation cycles.”

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**NEWS**

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**8 ^{TH} ONLINE VEDIC MATHEMATICS CONFERENCE**

This is scheduled for 12th, 13^{th} March 2022. Organised by the IAVM it features original research papers, ancient Indian mathematics, Vedic Maths in education plus workshops for teachers and students.

For more information see:

https://www.vedicmaths.org/community/calendar-of-events

or visit the IAVM website at https://instavm.org/

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**MATH2SHINE.COM**

Math2Shine is a Singapore-based Ed-Tech company co-founded by Lokesh Tayal and Kenneth Williams. It is promoting Vedic Mathematics globally.

Students can access over 1000 Vedic Maths, 2000 School Math, and 500 Abacus worksheets. Worksheets are automatically checked, and feedback is immediately provided. Math2Shine's worksheets generate new questions when retried. In addition to worksheets, students have access to hundreds of videos and Kenneth Williams e-books.

Math2Shine provides a Tutor Portal. It offers many benefits to the tutor community.

- Tutors become more productive and follow best teaching practices.
- Tutors can see students work in real-time and provide analytics-driven feedback.
- Teachers can assign classwork and homework with a few clicks.
- Tutors do not have to do the administrative tasks of designing and checking worksheets.

Math2Shine is looking for a Tutors association. Vedic Math, School Math, and Abacus tutors would significantly benefit from the Math2Shine platform. Math2Shine is also looking for tutors to translate Math2Shine into other languages.

You can contact Math2Shine at or to explore the opportunities to teach and spread Vedic Math globally.

For more insight into Math2Shine see: www.math2shine.com / https://www.youtube.com/user/math2shine/videos.

**VE****DIC MATHEMATICS ACADEMY COURSES**

The next Teacher Training Course starts on 14^{th} March 2022 on the Math2Shine platform.

For details of this and the several other courses offered please see:

https://courses.vedicmaths.org/

We will start taking registrations shortly.

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**ARTICLE FOR NEWSLETTER 132**

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**OSCULATION MAGIC**

Tirthaji’s osculation technique has many applications: in divisibility testing, converting fractions to decimals, multiplications, divisions, checking calculations, continued fractions, number bases, substituting etc.

The osculators are explained in Tirthaji’s book (chapters 29 and 30) and here we look at how to generate the osculators in sequence from the first one.

In many of the applications we need to select a suitable osculator from the many that can be used.

For example, if we need to test if a certain number is divisible by 23, Tirthaji’s osculation technique allows us to work 2 figures at a time, while dividing by 3, if we know that **299** is a multiple of 23.

Or, we can work 6 digits at a time, dividing by 4, if we know that **3999999** is a multiple of 23.

How can we find these convenient multiples?

In fact, for whole numbers, D, that end in 1, 3, 7 or 9 we can always get a multiple that ends in one 9, two 9s, three 9s, or indeed any number of 9s.

There will even be some which consists entirely of a series of 9s.

In Tirthaji’s osculation technique we begin by finding a multiple of D that ends in 9. So in the case of D = 23 we multiply 23 by 3 to get 69. The osculator is then 7: *one more than the number before* the 9. We can say that if D = 23 then P = 7, where P is called the osculator.

For the number 23, the multiples that end in 1, 2, 3 etc. 9s are:

69, 299. 20999, 89999, 1699999, 3999999 and so on.

The osculators for these – *one more than the number before* the 9s are:

**7**, 3, 21, 9, 17, 4 etc.

Now, **these can be obtained by repeatedly osculating by the ****1 ^{st} osculator **

**(**

**7**

**in this example).**

Osculation consists of multiplying the last digit of a number by the osculator,

then adding on any other digits and

subtracting D if necessary.

To make this clear let us generate the osculators for D = 23. Those osculators are **7**, **3**, **21**, **9**, **17**, **4** etc.

This is how it is done…

**7**×**7** = 49 and 49 – 2×23 = **3**; next we osculate this 3 with 7

3×**7** = **21**; next we osculate this 21 with 7

1×**7** + 2 = **9**;

9×**7** = 63 and 63 – 2×23 = **17**;

7×**7** + 1 = 50 and 50 – 2×23 =** 4**.

And so on.

This means we can easily generate the osculators from the first one by repeated osculation. We just osculate until we get a suitable one that we can use.

Let us take another example: D = 49.

The first osculator is 5 (one more than 4).

Now osculate repeatedly with 5, casting out any 49s as needed: we get **5**, **25**, **27**, **37**, **38** etc. That is:

**5**×**5** = **25**;

5×**5** + 2= **27**;

7×**5** + 2 = **37**;

7×**5** + 3 = **38**;

8×**5** + 3 = **43**.

And so on. (No casting out of 49s is necessary up to here.)

This tells us that 49, 2499, 26999, 369999 etc. are multiples of 49.

The number of 49s contained in each of these multiples is also easily available by an application of the Sutra *The Remainders By the Last Digit*.

For D = 49 there are 42 osculators before they start to repeat (though we can generate multiples with any number of 9s at the end!), and we can generate them from right to left if we wish.

We can also jump forwards or backwards through the osculators, getting every 2^{nd}, 3^{rd}, 4^{th} etc. one. And we can get specific osculators as required, or find the number of osculators in the cycle (which is also the number of digits in the recurring decimal for n/D).

There are many other beautiful and astounding properties within these osculation cycles.

End of article.

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Editor: Kenneth Williams

The Vedic Mathematics web site is at: https://www.vedicmaths.org

22 – 02 – 2022