# 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?**

- Take the last digit of the number,
**3**, and multiply it by the**Ekadhika**,**5**, to get**15**. - Add this
**15**to the remaining number,**83**to get**98**. - Remove the last digit,
**8**, from**98**and multiply it by**5**to get**40.** - 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**.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s **30MSC 2023 #21: Counting**