Divisibility Rules: A Complete Guide with Examples
In the world of mathematics, divisibility rules are a handy set of shortcuts that help determine if a number is divisible by another number without actually performing the division. These rules are particularly useful for simplifying calculations, identifying prime numbers, and understanding number properties.
Divisibility Rules Explained
Here's a breakdown of divisibility rules for numbers 2 through 13:
Divisible by | Rule | Example |
---|---|---|
2 | A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8). | 124 is divisible by 2 because 4 is even. |
3 | A number is divisible by 3 if the sum of its digits is divisible by 3. | 123 is divisible by 3 because 1 + 2 + 3 = 6, which is divisible by 3. |
4 | A number is divisible by 4 if the last two digits are divisible by 4. | 240 is divisible by 4 because 40 is divisible by 4. |
5 | A number is divisible by 5 if the last digit is 0 or 5. | 325 is divisible by 5 because the last digit is 5. |
6 | A number is divisible by 6 if it is divisible by both 2 and 3. | 108 is divisible by 6 because it is divisible by both 2 and 3. |
7 | Double the last digit, subtract it from the remaining number, and check if the result is divisible by 7. If it is, the original number is also divisible by 7. | 343 is divisible by 7 because (34 - (3 * 2)) = 28, which is divisible by 7. |
8 | A number is divisible by 8 if the last three digits are divisible by 8. | 1024 is divisible by 8 because 24 is divisible by 8. |
9 | A number is divisible by 9 if the sum of its digits is divisible by 9. | 729 is divisible by 9 because 7 + 2 + 9 = 18, which is divisible by 9. |
10 | A number is divisible by 10 if the last digit is 0. | 560 is divisible by 10 because the last digit is 0. |
11 | A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or divisible by 11. | 121 is divisible by 11 because (1 + 1) - 2 = 0. |
12 | A number is divisible by 12 if it is divisible by both 3 and 4. | 240 is divisible by 12 because it is divisible by both 3 and 4. |
13 | Subtract 4 times the last digit from the remaining number, and check if the result is divisible by 13. If it is, the original number is also divisible by 13. | 260 is divisible by 13 because (26 - (4 * 0)) = 26, which is divisible by 13. |
Applications of Divisibility Rules
Divisibility rules have various applications in mathematics and everyday life:
- Simplifying Calculations: They can speed up calculations by quickly determining if a number is divisible by another number without performing long division.
- Prime Number Identification: Divisibility rules help identify prime numbers, which are numbers greater than 1 that are only divisible by 1 and themselves.
- Factoring Numbers: Knowing divisibility rules helps factor numbers into their prime factors.
- Problem Solving: They are useful in solving mathematical problems related to numbers and their properties.
Example: Finding Prime Numbers
Let's use divisibility rules to identify prime numbers between 1 and 10. We can quickly eliminate numbers that are divisible by 2, 3, 5, or 7.
- 2, 3, 5, 7 are prime numbers.
- 4, 6, 8, 9, 10 are not prime numbers because they are divisible by numbers other than 1 and themselves.
Conclusion
Understanding divisibility rules is an essential skill in mathematics. They provide a foundation for simplifying calculations, identifying prime numbers, and exploring number properties. By mastering these simple rules, you can enhance your understanding of numbers and improve your problem-solving abilities.