# Working of divisibility for 3 and 9

We are aware that when any number has its unit digit as 0, 2, 4, 6, or 8 (all even numbers), the number is said to be divisible by 2. Similarly, to check if a number is divisible by 3 or 9, we simply sum up the digits of the number and see if it’s divisible by 3 or 9. If yes, then the number is divisible by 3 or 9 respectively.

For example, let’s take a random number N = 45831, and see if it’s divisible by 3 or not.

Divisibility is when you can divide the number evenly by a factor, leaving no remainder. If we take a real life example, a pizza having 4 slices, can be evenly distributed between 2 people, while it cannot be evenly given between 3 people. Here we say, 4 is divisible by 2. Divisibility rules help you work with large numbers to find their factors, and determine if the numbers are prime or composite.

## Ever wondered how and why the divisibility formula works?

Well, I didn’t until I read about it recently. Let me share with you the working behind the strategy. As you read further, you’ll see that the propositions used, are all based on number theories, arithmetic laws & properties.

We start with basic representation of numbers. Consider a number 4708492, the 3 common ways of expressing the number are:

• Standard form: 4,708,492
• Expanded form: 4 x (1,000,000) + 7 x (100,000) + 0 x (10,000) + 8 x (1000) + 4 x (100) + 9 x (10) + 2 x (1)
• Words form: Four million, seven hundred eight thousand, four hundred, and ninety two

Through the above equation, theories conclude that for N to be divisible by 3, if (a + b + c + d + e) is divisible by 3 then, N is divisible by 3. Same logic applies for divisibility by 9!

Let’s take some examples for both 3 and 9 and see how the equation unfolds…

We saw the working behind the divisibility for 3 and 9, similarly, there exists respective reasoning for other divisibility checks as well. The more we dig into the nuances of math concepts, the more magical it appears.

1. Nithya Veena says:

Nicely explained.

Liked by 1 person

2. Math Sux says:

Interesting! Thank you for this! 🙂
https://mathsux.org/

Liked by 1 person

1. Aswini Sekar says:

Thank you..

Liked by 1 person

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