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30MSC 2023 #20: Divisibility

30MSC 2023 #20: Divisibility

There are simple divisibility rules for some numbers. The sutra, Only the Last Terms  is used to determine the divisibility by 2, 4, 5, 8 and 10. Digit sums are used to determine divisibility by 3 and 9. For composite numbers like 15, for example, the divisibility rules for its factors 3 and 5 must be applied.

There are no easy divisibility rules for other prime numbers like 7, 13, 17, 19 etc. in our conventional system but in Vedic Math, we have Osculation.

There are positive and negative osculators.  The positive osculators are simply the Ekadhika of the numbers: for 19, it is 2; for 29, 39 and 49, they are 3, 4 and 5 respectively. Numbers not ending in 9 can be multiplied by a suitable factor to make the product end in 9. 13 can be multiplied by 3 to get 39; thus the positive osculator of 13 is 4.

Our featured question requires the osculator for 7. We can multiply it by 7 to make it 49. Therefore, the Osculator for 7 is the Ekadhika of 49, which is 5.

Let us take  simple example to show how osculation operates: is 833 exactly divisible by 7?

  1. Take the last digit of the number, 3,  and multiply it by the Ekadhika, 5, to get 15.
  2. Add this 15 to the remaining number, 83 to get 98.
  3. Remove the last digit,8, from 98 and multiply it by 5 to get 40.
  4. Add 40 to 9 to get 49 which is divisible by 9. So 833 is divisible by 7.  

Note that if we cannot recognize 49 as divisible by 7, we can continue osculating. Multiply 9 by 5 and get 45. Then if it is added to 4, we will again get 49. Arriving at the same figure means divisibility.

Divisibility rules are discussed extensively in  Chapter 10 of our forthcoming book, 30 Master Strategies in Computing Vol II.

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