Mathematics can often seem daunting, but breaking down concepts into manageable parts can make it much more accessible. One such concept is the divisibility rule of 6. This rule is a simple yet powerful tool that allows you to determine whether a number is divisible by 6 without having to perform the actual division. In this article, we’ll explore this rule in detail, provide clear examples, and explain how you can apply it to everyday math problems.
What is the Divisibility Rule of 6?
The divisibility rule of 6 states that a number is divisible by 6 if it meets two specific criteria:
- The number is divisible by 2.
- The number is divisible by 3.
This means that for a number to be divisible by 6, it must be both even (divisible by 2) and have a sum of digits that is divisible by 3. Let’s dive into each of these criteria in more detail.
Criteria 1: Divisibility by 2
A number is divisible by 2 if its last digit is an even number. The even numbers are 0, 2, 4, 6, and 8. If the last digit of a number is one of these, then the number is even and therefore divisible by 2.
Examples of Numbers Divisible by 2:
- 18: The last digit is 8, which is even.
- 24: The last digit is 4, which is even.
- 54: The last digit is 4, which is even.
If a number does not end in an even digit, it is not divisible by 2 and, consequently, cannot be divisible by 6.
Criteria 2: Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. To determine this, add up all the digits in the number and check if the resulting sum is a multiple of 3.
Examples of Numbers Divisible by 3:
- 18: The sum of the digits is 1 + 8 = 9, which is divisible by 3.
- 24: The sum of the digits is 2 + 4 = 6, which is divisible by 3.
- 54: The sum of the digits is 5 + 4 = 9, which is divisible by 3.
If the sum of the digits is not divisible by 3, the number itself is not divisible by 3 and, therefore, cannot be divisible by 6.
Applying the Divisibility Rule of 6
Now that we understand the two criteria, let’s see how they work together to determine if a number is divisible by 6. A number must pass both tests: it must be even and the sum of its digits must be divisible by 3. Let’s look at some examples to clarify this.
Example 1: 18
Step 1: Check Divisibility by 2
- The last digit of 18 is 8, which is even. So, 18 is divisible by 2.
Step 2: Check Divisibility by 3
- Sum of the digits: 1 + 8 = 9.
- Since 9 is divisible by 3, 18 is also divisible by 3.
Conclusion: 18 is divisible by 6 (since it meets both criteria).
Example 2: 24
Step 1: Check Divisibility by 2
- The last digit of 24 is 4, which is even. So, 24 is divisible by 2.
Step 2: Check Divisibility by 3
- Sum of the digits: 2 + 4 = 6.
- Since 6 is divisible by 3, 24 is also divisible by 3.
Conclusion: 24 is divisible by 6 (since it meets both criteria).
Example 3: 35
Step 1: Check Divisibility by 2
- The last digit of 35 is 5, which is odd. So, 35 is not divisible by 2.
Conclusion: Since 35 is not divisible by 2, it is not divisible by 6. We don’t need to check the divisibility by 3.
Example 4: 54
Step 1: Check Divisibility by 2
- The last digit of 54 is 4, which is even. So, 54 is divisible by 2.
Step 2: Check Divisibility by 3
- Sum of the digits: 5 + 4 = 9.
- Since 9 is divisible by 3, 54 is also divisible by 3.
Conclusion: 54 is divisible by 6 (since it meets both criteria).
Why the Divisibility Rule of 6 is Useful
The divisibility rule of 6 is a quick and efficient way to determine if a number can be divided by 6 without performing the actual division. This can save time and effort, especially when dealing with large numbers or multiple calculations. Here are some scenarios where this rule can be particularly useful:
- Simplifying Fractions: When simplifying fractions, knowing if the numerator or denominator is divisible by 6 can help in reducing the fraction more efficiently.
- Math Problems and Puzzles: The rule can be handy for solving various math problems and puzzles that involve divisibility.
- Financial Calculations: In financial contexts, such as budgeting or accounting, quickly determining divisibility can help in allocating resources or balancing books.
Practice Makes Perfect
The best way to master the divisibility rule of 6 is through practice. Here are a few numbers to test your understanding:
- 36: Is it divisible by 6?
- Check divisibility by 2: Last digit is 6 (even), so yes.
- Check divisibility by 3: Sum of digits is 3 + 6 = 9 (divisible by 3), so yes.
- Conclusion: 36 is divisible by 6.
- 50: Is it divisible by 6?
- Check divisibility by 2: Last digit is 0 (even), so yes.
- Check divisibility by 3: Sum of digits is 5 + 0 = 5 (not divisible by 3), so no.
- Conclusion: 50 is not divisible by 6.
- 72: Is it divisible by 6?
- Check divisibility by 2: Last digit is 2 (even), so yes.
- Check divisibility by 3: Sum of digits is 7 + 2 = 9 (divisible by 3), so yes.
- Conclusion: 72 is divisible by 6.
- 91: Is it divisible by 6?
- Check divisibility by 2: Last digit is 1 (odd), so no.
- Conclusion: 91 is not divisible by 6.
By practicing with different numbers, you’ll become more comfortable and quicker at applying the divisibility rule of 6.
Conclusion
The divisibility rule of 6 is a straightforward yet powerful tool that can make certain math problems much easier to handle. By ensuring a number is divisible by both 2 and 3, you can quickly determine its divisibility by 6. Remember, practice is key to mastering this rule, so keep testing yourself with different numbers. Whether you’re simplifying fractions, solving math puzzles, or working on financial calculations, this rule will serve you well