The divisibility rule for 6 states that a number is divisible by 6 if it is divisible by both 2 and 3, meaning the number must be even (last digit 0, 2, 4, 6, or 8) and the sum of its digits must be divisible by 3. This rule integrates the principles of addition, subtraction, multiplication, and division, providing a streamlined approach to evaluating divisibility among integers, a key subset of rational numbers. In algebra, understanding and applying the divisibility rules enhances problem-solving efficiency and simplifies equation manipulation. Since divisibility concerns only integers, this rule does not apply to irrational numbers, which cannot be expressed as fractions or ratios. Mastery of this rule is essential in foundational mathematics, facilitating the efficient breakdown and evaluation of numbers in both educational and practical contexts.
Download Proof of Divisibility Rule of 6 in PDF
What is the Divisibility Rule of 6?
Proof of Divisibility Rule of 6
Download Proof of Divisibility Rule of 6 in PDF
Let’s use the number 2436 to demonstrate the proof of the divisibility rule of 6 step-by-step:
Step 1: Consider the number 2436.
Step 2: Check for divisibility by 2 (evenness).
- Number 2436 is even (last digit is 6), so it passes the first criterion of divisibility by 2.
Step 3: Check for divisibility by 3.
- The sum of the digits of 2436 is calculated as follows:2+4+3+62+4+3+6
Step 4: Calculate the sum of the digits.
- Adding the digits together:2+4+3+6 = 15
Step 5: Check if the sum from Step 4 is divisible by 3.
- Since 15 is divisible by 3 (15 ÷ 3 = 5), the number meets the second criterion.
Step 6: Apply the divisibility rule of 6.
- Since 2436 is divisible by both 2 and 3, according to our checks in Steps 2 and 5, it must be divisible by 6.
Step 7: Conclusion from divisibility checks.
- Both conditions (divisibility by 2 and 3) are satisfied, confirming that 2436 is divisible by 6.
Step 8: Summarize the proof of divisibility by 6.
- The number 2436 is divisible by 6 because it is even (satisfying divisibility by 2) and the sum of its digits (15) is divisible by 3. Thus, the divisibility rule of 6 is confirmed for 2436.
Divisibility Rule of 6 and 7
Divisibility Rule of 6
To determine whether a number is divisible by 6, it must satisfy two conditions: the number must be divisible by both 2 and 3. Here is a simple step-by-step process to check divisibility by 6:
- Even Number Check: First, ensure the number is even. This can be quickly verified if the last digit of the number is 0, 2, 4, 6, or 8.
- Sum of Digits: Next, add up all the digits of the number. If the resulting sum is divisible by 3 (i.e., the sum is 3, 6, 9, 12, etc.), then the number meets the second criterion.
Example: Consider the number 132:
- It ends in 2, which is even, satisfying the first condition.
- The sum of the digits is 1+3+2 = 6, which is divisible by 3, satisfying the second condition. Thus, 132 is divisible by 6.
Divisibility Rule of 7
The rule for checking divisibility by 7 is a bit more complex than for most other single-digit numbers. Here is a methodical way to determine if a number is divisible by 7:
- Last Digit Times Two: Take the last digit of the number, double it.
- Subtract from the Rest: Subtract this doubled figure from the rest of the number (i.e., the number without its last digit).
- Result: If the result is 0 or divisible by 7, then the original number is divisible by 7. This process can be repeated if the new number is still too large.
Example: Consider the number 161:
- The last digit is 1. Doubling 1 gives 2.
- Subtract 2 from 16 (the remaining part of the number) to get 14.
- Since 14 is divisible by 7, 161 is also divisible by 7.
Divisibility Rule of 6 and 9
Divisibility Rule of 6
The divisibility rule for 6 requires that a number must be divisible by both 2 and 3. Here’s how you can easily determine if a number is divisible by 6:
- Check for Evenness: First, ensure the number is even. This means the last digit should be 0, 2, 4, 6, or 8.
- Sum of Digits: Calculate the sum of all the digits in the number. If this sum is divisible by 3 (i.e., the sum is 3, 6, 9, 12, etc.), then the number also meets the second requirement.
Example: Consider the number 234:
- It ends in 4, which is even.
- The sum of the digits is 2+3+4=9, which is divisible by 3. Therefore, 234 is divisible by 6.
Divisibility Rule of 9
The divisibility rule for 9 is straightforward and involves only the sum of the digits:
- Sum of Digits: Add up all the digits in the number.
- Check Divisibility: If the resulting sum is divisible by 9, then so is the entire number.
Example: Consider the number 729:
- The sum of the digits is 7+2+9 = 18.
- Since 18 is divisible by 9, the number 729 is also divisible by 9.
Divisibility Rule of 6 Examples
Check if 258 is divisible by 6
Check for Evenness: The last digit of 258 is 8, which is an even number. This satisfies the first condition of being divisible by 2.
Sum of Digits: Calculate the sum of the digits in 258.
2+5+8 = 15
Check Divisibility by 3: The sum, 15, is divisible by 3 (since 15÷3 = 5).
Conclusion: Since 258 meets both conditions, it is divisible by 6.
Check if 123 is divisible by 6
- Check for Evenness: The last digit of 123 is 3, which is not even. Therefore, it fails the divisibility by 2 test.
- Conclusion: Since it does not meet the first condition, there is no need to check the sum of its digits. 123 is not divisible by 6.
Check if 612 is divisible by 6
Check for Evenness: The last digit of 612 is 2, which is even.
Sum of Digits: Calculate the sum of the digits in 612.
6+1+2 = 9
Check Divisibility by 3: The sum, 9, is divisible by 3 (since 9÷3 = 3).
Conclusion: Since 612 meets both conditions, it is divisible by 6.
Check if 540 is divisible by 6
Check for Evenness: The last digit of 540 is 0, which is even.
Sum of Digits: Calculate the sum of the digits in 540.
5+4+0 = 9
Check Divisibility by 3: The sum, 9, is divisible by 3.
Conclusion: Since 540 meets both conditions, it is divisible by 6.
FAQs
A number must be even because 6 is a multiple of 2. Therefore, any number divisible by 6 must also be divisible by 2, which means its last digit must be even.
To determine if a number is divisible by 3, sum up all its digits. If the result is divisible by 3 (such as 3, 6, 9, 12, etc.), then the original number is also divisible by 3.
Yes, consider the number 14. It is even (last digit is 4), but the sum of its digits (1 + 4 = 5) is not divisible by 3. Thus, 14 is not divisible by 6.
The smallest number divisible by 6 is 6 itself.
Yes, the divisibility rules apply to negative numbers as well. A negative number is divisible by 6 if, when considered without its negative sign, it fulfills the conditions of being divisible by both 2 and 3.
Understanding the divisibility rule of 6 is helpful in simplifying fractions, finding common denominators, and solving problems that require factorization or checking for multiples.
Yes, it helps in determining whether 6 is a factor of a number, which is crucial in breaking down a number into its prime factors.
While both numbers require the target number to be even, for divisibility by 12, the sum of the digits must be divisible by 3 and the number formed by the last two digits must also be divisible by 4.
No, there are no exceptions. If a number meets the conditions of being even and having digits that sum to a multiple of 3, it is always divisible by 6.
