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.