How to Determine if a Number is Divisible by 6: A Comprehensive Guide

Divisibility is a fundamental concept in mathematics, and understanding how to determine if a number is divisible by 6 is crucial for both educational purposes and practical applications. This guide provides a clear and concise method to check the divisibility of a number by 6, along with examples and a comparison of different methods to ensure accurate results.

Criteria for Divisibility by 6

A number is divisible by 6 if and only if it is divisible by both 2 and 3. This requires checking two conditions:

Divisibility by 2:

A number is divisible by 2 if its last digit is even. The possible even digits are 0, 2, 4, 6, and 8. For example, consider the number 24:

Check the last digit: The last digit is 4, which is even.

Divisibility by 3:

A number is divisible by 3 if the sum of its digits is divisible by 3. For the number 24:

Calculate the sum of the digits: 2 4 6. Check divisibility: 6 is divisible by 3.

Since both conditions are satisfied, 24 is divisible by 6.

Simplified Method Using the Last Digit

For a more efficient approach, if a number is even, you can divide it by 2 and obtain a new number N2. Then, you can check if this new number is divisible by 3 using a simple digit-sum method. Here's the step-by-step process:

Divide the number by 2 to obtain N2. Sum the digits of N2 until a single-digit result is obtained. If the final single-digit result is 3, 6, or 9, then the original number was divisible by 6.

Example: Check if 126 is divisible by 6 using this method:

Divide 126 by 2: 126 / 2 63. Sum the digits of 63: 6 3 9. 9 is divisible by 3, so 126 is divisible by 6.

Practical Examples

Let's explore a more complex example with a large number, 2130220413208008557448267936706.

Method 1: Using Sum of Digits

Check if the number ends in an even digit and if the sum of its digits is divisible by 3:

Last digit is even: 6 (even). Calculate the sum of digits: 2 1 3 0 2 2 0 4 1 3 2 0 8 0 0 8 5 5 7 4 4 8 2 6 7 9 3 6 7 0 6 115. 115 is not divisible by 3.

Conclusion: The number 2130220413208008557448267936706 is not divisible by 6.

Method 2: Using Remainder

Divide the number by 6 and check the remainder:

2130220413208008557448267936706 / 6 quotient with remainder 4.

Conclusion: The number is not divisible by 6.

Method 3: Using a Calculator

Divide the number by 6 using a calculator and check the remainder:

The result is 355003402201334759574711322784.33333333333333333333. The decimal part is not zero, indicating a remainder.