Understanding the Concept of Divisibility Rules
Divisibility rules are shortcuts that allow us to quickly determine whether one number can be divided by another without leaving a remainder. These rules simplify mathematical calculations by providing simple tests that can be applied to a number to check for divisibility. Each rule is tailored to a specific divisor, making it possible to evaluate large numbers without performing long division.
The primary concept behind divisibility rules is to use the properties of numbers and their digits to identify factors. For instance, some rules focus on the last digit of a number, while others consider the sum of the digits or other operations. By applying these straightforward tests, you can save time and effort in various mathematical tasks, from simplifying fractions to solving complex equations.
Divisibility rules are not only useful for quick calculations but also provide deeper insights into the structure and properties of numbers. Understanding these rules enhances your number sense and helps you recognize patterns, making it easier to work with numbers in everyday life and advanced mathematics.
In this guide, we will explore the divisibility rules for the numbers 2, 3, 5, 6, 7, 8, 9, 11, and 12. We will explain each rule, provide clear examples, and show you how to apply them effectively.
Divisibility Rule for 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. This rule is straightforward and easy to apply.
Examples:
- 24: The last digit is 4, which is even. Thus, 24 is divisible by 2.
- 137: The last digit is 7, which is odd. Therefore, 137 is not divisible by 2.
Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3. To apply this rule, simply add all the digits of the number and check the sum.
Examples:
- 123: Sum of digits is 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is divisible by 3.
- 85: Sum of digits is 8 + 5 = 13. Since 13 is not divisible by 3, 85 is not divisible by 3.
Divisibility Rule for 5
A number is divisible by 5 if its last digit is either 0 or 5. This rule is very easy to remember and apply.
Examples:
- 45: The last digit is 5. Thus, 45 is divisible by 5.
- 62: The last digit is 2. Therefore, 62 is not divisible by 5.
Divisibility Rule for 6
A number is divisible by 6 if it meets two criteria:
- It is divisible by 2.
- It is divisible by 3.
Examples:
- 36: The last digit is 6 (even), so it is divisible by 2. Sum of digits is 3 + 6 = 9, which is divisible by 3. Thus, 36 is divisible by 6.
- 25: The last digit is 5 (odd), so it is not divisible by 2. Therefore, 25 is not divisible by 6.
Divisibility Rule for 7
A number is divisible by 7 if, after doubling the last digit and subtracting it from the rest of the number, the result is divisible by 7. This rule can be a bit more complex to apply, but with practice, it becomes easier.
Examples:
- 203: Double the last digit (3) to get 6. Subtract 6 from the rest of the number (20) to get 14, which is divisible by 7. Therefore, 203 is divisible by 7.
- 273: Double the last digit (3) to get 6. Subtract 6 from the rest of the number (27) to get 21, which is divisible by 7. Thus, 273 is divisible by 7.
Divisibility Rule for 8
A number is divisible by 8 if the last three digits of the number are divisible by 8. This rule is particularly useful for larger numbers.
Examples:
- 1048: The last three digits are 048. Since 48 is divisible by 8, 1048 is divisible by 8.
- 1234: The last three digits are 234. Since 234 is not divisible by 8, 1234 is not divisible by 8.
Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9. This rule is similar to the rule for 3 but with a different divisor.
Examples:
- 729: Sum of digits is 7 + 2 + 9 = 18. Since 18 is divisible by 9, 729 is divisible by 9.
- 623: Sum of digits is 6 + 2 + 3 = 11. Since 11 is not divisible by 9, 623 is not divisible by 9.
Divisibility Rule for 11
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11. This rule can be more complex but is effective.
Examples:
- 121: The sum of digits in odd positions is 1 + 1 = 2. The sum of digits in even positions is 2. The difference is 2 – 2 = 0, which is divisible by 11. Therefore, 121 is divisible by 11.
- 3456: The sum of digits in odd positions is 3 + 5 = 8. The sum of digits in even positions is 4 + 6 = 10. The difference is 8 – 10 = -2, which is not divisible by 11. Thus, 3456 is not divisible by 11.
Divisibility Rule for 12
A number is divisible by 12 if it is divisible by both 3 and 4. This rule combines the rules for these two divisors.
Examples:
- 48: Sum of digits is 4 + 8 = 12, which is divisible by 3. The last two digits are 48, which is divisible by 4. Therefore, 48 is divisible by 12.
- 60: Sum of digits is 6 + 0 = 6, which is divisible by 3. The last two digits are 60, which is divisible by 4. Thus, 60 is divisible by 12.
Summary
Understanding and applying divisibility rules can greatly simplify many mathematical tasks, from basic arithmetic to more complex problem-solving. Here’s a quick summary of the divisibility rules we covered:
- 2: A number is divisible by 2 if its last digit is even.
- 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- 5: A number is divisible by 5 if its last digit is 0 or 5.
- 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- 7: A number is divisible by 7 if doubling the last digit and subtracting it from the rest of the number yields a result divisible by 7.
- 8: A number is divisible by 8 if the last three digits are divisible by 8.
- 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- 11: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11.
- 12: A number is divisible by 12 if it is divisible by both 3 and 4.
By mastering these rules, you’ll be equipped with powerful tools to handle a wide range of mathematical problems more efficiently. Keep practicing with different numbers, and soon these rules will become second nature