Master Divisibility Rules: Easy Guide with Printable Worksheets & Word Problems

Introduction to Divisibility Rules

Divisibility rules are like shortcuts that help you quickly determine whether one number can be evenly divided by another. Instead of performing complex division, these handy rules allow you to work smarter, not harder, when solving math problems. From grade school to advanced math, divisibility plays a crucial role in problem-solving and number theory. Mastering these rules will make large numbers less intimidating and allow you to work through math with greater ease.

Why Learning Divisibility Rules is Essential

Understanding divisibility rules is key to making math simpler and more manageable. They serve as the foundation for many other mathematical concepts, such as factoring and simplifying fractions. Plus, these rules are not just confined to classrooms—they pop up in real-life situations like figuring out bill splits, measuring ingredients in recipes, and calculating taxes.

Mastering divisibility rules also prepares you for more advanced topics in math like algebra, number theory, and geometry, and helps you tackle standardized tests with confidence.

The Basics of Divisibility

Before we dive into specific rules, let’s start with some basics. A number is divisible by another if, after dividing, the result is a whole number with no remainder. For example, 12 is divisible by 3 because 12 ÷ 3 equals 4, a whole number. On the other hand, 13 ÷ 3 leaves a remainder, so 13 is not divisible by 3.

Divisibility Rule for 2

This is one of the easiest divisibility rules to remember. Any number that ends in 0, 2, 4, 6, or 8 is divisible by 2. In other words, if a number is even, it’s divisible by 2.

Example:

• 24 ends in 4, so it is divisible by 2.
• 37 ends in 7, which is odd, so it is not divisible by 2.

Divisibility Rule for 3

To check if a number is divisible by 3, sum all of its digits. If the result is divisible by 3, then the original number is also divisible by 3.

Example:

• For 123, the sum of the digits is 1 + 2 + 3 = 6, and 6 is divisible by 3.
• For 145, the sum is 1 + 4 + 5 = 10, and 10 is not divisible by 3.

Divisibility Rule for 4

If the last two digits of a number are divisible by 4, then the entire number is divisible by 4. This rule is handy when dealing with large numbers.

Example:

• For 112, the last two digits (12) are divisible by 4, so 112 is divisible by 4.
• For 157, the last two digits (57) are not divisible by 4.

Divisibility Rule for 5

The rule for 5 is probably the simplest. If a number ends in 0 or 5, it’s divisible by 5.

Example:

• 25 ends in 5, so it is divisible by 5.
• 18 ends in 8, so it is not divisible by 5.

Divisibility Rule for 6

A number is divisible by 6 if it follows the rules for both 2 and 3. In other words, the number must be even, and the sum of its digits must be divisible by 3.

Example:

• 72 is divisible by both 2 (because it’s even) and 3 (7 + 2 = 9, and 9 is divisible by 3).
• 54 is divisible by 3 but not by 2, so it is not divisible by 6.

Divisibility Rule for 7

The rule for 7 is a bit tricky. Take the last digit of the number, double it, and subtract it from the remaining digits. If the result is divisible by 7, so is the original number.

Example:

• For 672, double the last digit (2 × 2 = 4), and subtract that from 67 (67 − 4 = 63). Since 63 is divisible by 7, 672 is divisible by 7.

Divisibility Rule for 8

To check divisibility by 8, look at the last three digits of the number. If those digits are divisible by 8, then the entire number is divisible by 8.

Example:

• For 1,536, the last three digits (536) are divisible by 8, so 1,536 is divisible by 8.
• For 1,527, the last three digits (527) are not divisible by 8.

Divisibility Rule for 9

The rule for 9 is similar to the rule for 3. Add all the digits of the number, and if the sum is divisible by 9, then the number itself is divisible by 9.

Example:

• For 729, the sum of the digits is 7 + 2 + 9 = 18, and 18 is divisible by 9.
• For 733, the sum is 7 + 3 + 3 = 13, and 13 is not divisible by 9.

Divisibility Rule for 10

This one is straightforward. If a number ends in 0, it’s divisible by 10.

Example:

• 130 ends in 0, so it is divisible by 10.
• 137 does not end in 0, so it is not divisible by 10.

Printable Worksheets for Divisibility Rules

Practicing divisibility rules can be made easy with printable worksheets. These worksheets provide exercises with step-by-step solutions, helping students build confidence in their divisibility skills. You can find high-quality, downloadable worksheets online to reinforce your learning.

Word Problems to Strengthen Understanding

Word problems are a great way to apply divisibility rules in real-life scenarios. They help sharpen your critical thinking and problem-solving skills.

Sample Problem:

• Sarah has 48 apples and wants to divide them equally among her 6 friends. How many apples will each friend get?

Answer: Divide 48 by 6. Each friend will get 8 apples.

Conclusion

Mastering divisibility rules doesn’t have to be difficult. With a little practice, you’ll find that these rules can save time and simplify many mathematical problems. They’re foundational skills that will serve you well, whether in school, work, or daily life.

FAQs

Can I learn divisibility rules without a tutor?

Absolutely! With the help of printable worksheets, online resources, and regular practice, anyone can master divisibility rules on their own.

What are the hardest divisibility rules to understand?
The rule for 7 tends to be the trickiest because of its multi-step process.

How can I practice divisibility in everyday life?
You can apply divisibility rules when splitting bills, dividing groups evenly, or calculating discounts while shopping.

Are there any apps for learning divisibility?
Yes, several educational apps like Khan Academy and Mathway offer exercises and quizzes on divisibility rules.

Why are divisibility rules essential for competitive exams?
Competitive exams often include quick calculations, and divisibility rules help you solve problems faster without using a calculator.