Proving the Divisibility Rule for 3 Using Vedic Mathematics Techniques

By Dr. Yogesh Chandna - Author of 'Vedic Mathematics Step by Step'

In this article, we will explore the mathematical proof behind the divisibility rule for 3. This rule states that if the sum of the digits of a number is divisible by 3, then the number itself is also divisible by 3. We will delve into the logic and steps required to understand and prove this rule, leveraging Vedic Mathematics techniques.

Understanding the Problem

Consider an n-digit number represented as x[n]x[n-1]…x[1], where x[n] is the n-th digit. This number can be mathematically expressed as:

$$ sum_{i1}^{n} x_i cdot 10^{i-1} $$

For example, the number 372 can be broken down as:

372 3 * 10^2 7 * 10^1 2 * 10^0 300 70 2

Key Mathematical Proof

To understand the proof, we first rewrite the number in a different form:

$$ sum_{i1}^{n} x_i cdot 10^{i-1} sum_{i1}^{n} x_i cdot (10^{i-1} - 1) sum_{i1}^{n} x_i $$

Breaking down the equation:

1. The first term, $sum_{i1}^{n} x_i cdot (10^{i-1} - 1)$, is divisible by 3 because $10^{i-1} - 1$ can be shown to be a multiple of 3 for all i from 1 to n, through mathematical induction. (Note that $10^i - 1 9k$ for some integer k.) 2. The second term, $sum_{i1}^{n} x_i$, is simply the sum of the digits of the number. 3. Since both terms are divisible by 3, the entire number is divisible by 3.

Example with a 5-Digit Number

Consider the 5-digit number 672. We can express it as:

672 6 * 10^2 7 * 10^1 2 * 10^0

This can also be broken down as:

672 6 * 999 70 2

Since 999 is a multiple of 3, and 70 2 is simply the sum of the digits, we can conclude that 672 is divisible by 3.

General Proof for Any Number

Extending this logic to any number of digits, if a number is expressed as:

10000a 1000b 100c 10d e

We can rewrite it as:

9999a 999b 99c 9d (a b c d e)

Here, the first part is clearly divisible by 3 since each term is a multiple of 3 or 9 (9k for some integer k). Therefore, for the entire number to be divisible by 3, the sum of the digits (second part) must also be divisible by 3.

Application of the Rule

Let's apply this to a 5-digit number, 12345:

1. Sum of the digits: 1 2 3 4 5 15 2. 15 is divisible by 3 3. Therefore, 12345 is divisible by 3

Note: This rule can be extended to numbers of any number of digits, and it also proves the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

By understanding and applying these principles, we can easily verify the divisibility of numbers using Vedic Mathematics techniques. This method not only simplifies the process of checking divisibility but also enhances our mathematical reasoning skills.