Divisibility Rules - Definition, Examples, Quiz, FAQ, Trivia
Learn math shortcuts for division with these easy-to-follow rules and patterns
What is Divisibility?
Divisibility means a number can be divided by another number without leaving any remainder.
For example, 10 is divisible by 5 because 10 ÷ 5 = 2 with no remainder. But 10 is not divisible by 3 because 10 ÷ 3 = 3 with a remainder of 1.
Divisibility rules are shortcuts that help us quickly figure out if one number can be divided evenly by another number, without actually doing the division. These rules are based on patterns in our number system.
Key Concept
If a number is divisible by another, when you divide them, you get a whole number with no remainder.
Divisibility Rules
Here are the most useful divisibility rules. You can use these to quickly check if a number is divisible by 2, 3, 5, 7, and other numbers:
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0,2,4,6,8) | 58 (ends with 8) |
| 3 | Sum of digits is divisible by 3 | 123 (1+2+3=6 ÷ 3=2) |
| 4 | Last two digits form a number divisible by 4 | 312 (12 ÷ 4=3) |
| 5 | Last digit is 0 or 5 | 75 (ends with 5) |
| 6 | Divisible by both 2 and 3 | 24 (even, 2+4=6 ÷3) |
| 7 | Double the last digit and subtract from the rest | 203 (20-6=14 ÷7) |
| 8 | Last three digits form a number divisible by 8 | 3,120 (120 ÷ 8=15) |
| 9 | Sum of digits is divisible by 9 | 288 (2+8+8=18 ÷9) |
| 10 | Ends with 0 | 150 (ends with 0) |
Remember
The rule for 7 is the trickiest! Practice with numbers like 21, 49, and 84 to get comfortable with it.
Practice Examples
Let's practice using the divisibility rules with some examples:
Example 1: Is 126 divisible by 3?
Solution: Add the digits: 1 + 2 + 6 = 9. Since 9 is divisible by 3, yes!
Example 2: Is 245 divisible by 5?
Solution: It ends with 5, so yes!
Example 3: Is 432 divisible by 4?
Solution: Look at the last two digits: 32. 32 ÷ 4 = 8, so yes!
Example 4: Is 343 divisible by 7?
Solution: Double the last digit (3×2=6), subtract from the rest: 34 - 6 = 28. Since 28 ÷ 7 = 4, yes!
Try these on your own:
1. Is 180 divisible by 6? (Hint: Check rules for 2 and 3)
2. Is 648 divisible by 9? (Add the digits)
3. Is 725 divisible by 5? (Look at the last digit)
Practice Tip
Start with smaller numbers and work your way up to larger ones as you get more comfortable with the rules.
Why Learn Divisibility Rules?
Building Number Sense: Divisibility rules help students understand how numbers are structured and how they relate to each other. This strengthens overall number sense - the intuitive understanding of numbers and their relationships.
Pattern Recognition: These rules are based on mathematical patterns. Learning them trains your brain to recognize patterns in numbers, which is a fundamental skill in all areas of mathematics.
Mental Math Skills: Divisibility rules allow you to quickly determine factors and multiples without performing lengthy division. This speeds up mental calculations and builds confidence with numbers.
Problem-Solving Foundation: Understanding divisibility is essential for working with fractions, simplifying expressions, finding common denominators, and solving more complex math problems.
Real-World Applications: Divisibility concepts are used in scheduling, computer programming, cryptography, and many everyday situations where we need to divide things equally.
Key Concepts Summary
- Divisibility means a number can be divided by another number with no remainder
- Divisibility rules are shortcuts to determine if division is possible without calculating
- For 2: Last digit must be even (0, 2, 4, 6, 8)
- For 3: Sum of all digits must be divisible by 3
- For 5: Last digit must be 0 or 5
- For 7: Double the last digit and subtract it from the remaining number - result must be divisible by 7
- For 10: Number must end with 0
- For 6: Number must satisfy rules for both 2 and 3
- For 9: Sum of all digits must be divisible by 9
- For 4: Last two digits must form a number divisible by 4
Divisibility Quiz
Test your knowledge with this 5-question quiz. Choose the correct answer for each question.
Math Trivia & Fun Facts
Discover interesting facts about numbers and divisibility:
Ancient Divisibility
Divisibility rules were first recorded by the ancient Greeks. Pythagoras and his followers were fascinated by the properties of numbers, especially divisibility.
Perfect Numbers
A perfect number equals the sum of its proper divisors. 6 is perfect because 1 + 2 + 3 = 6. The next perfect number is 28 (1+2+4+7+14=28).
Divisibility in Nature
Many flowers have petals in numbers divisible by certain patterns. Lilies have 3 petals, buttercups have 5, and daisies often have 21, 34, or 55 petals - all Fibonacci numbers!
Largest Prime
The largest known prime number has over 24 million digits! It was discovered in 2018 and is expressed as 2⁸²⁵⁸⁹⁹³³ - 1. Prime numbers are only divisible by 1 and themselves.
