Divisibility Rules Worksheet Pdf With Answers

Divisibility Rules Worksheet PDF with Answers: A Comprehensive Guide

Are you looking for a comprehensive resource to help you master divisibility rules? This article provides a detailed explanation of divisibility rules for various numbers, along with a sample worksheet (though you can't download a PDF here, I'll give you the content to create your own!) and answers. Mastering these rules will significantly improve your math skills and speed up your calculations. This guide is perfect for students of all levels, from elementary school to high school, and even for those brushing up on their math skills.

What are Divisibility Rules?

Divisibility rules are shortcuts to determine if a number is divisible by another number without performing long division. They are based on patterns in the number system and can save you considerable time and effort when working with larger numbers. Understanding these rules is crucial for simplifying fractions, finding factors, and solving various mathematical problems.

Divisibility Rules for Common Numbers

Let's explore the divisibility rules for some of the most frequently encountered numbers:

2: Even Numbers

A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).

• Example: 124 is divisible by 2 because its last digit, 4, is even. 375 is not divisible by 2 because its last digit, 5, is odd.

3: Sum of Digits

A number is divisible by 3 if the sum of its digits is divisible by 3.

• Example: Consider the number 123. 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is divisible by 3. Let's try 456: 4 + 5 + 6 = 15. 15 is divisible by 3, therefore 456 is also divisible by 3.

4: Last Two Digits

A number is divisible by 4 if its last two digits form a number divisible by 4.

• Example: Consider the number 1312. The last two digits are 12, and 12 is divisible by 4 (12/4 = 3). Therefore, 1312 is divisible by 4. However, 2357 is not divisible by 4 because 57 is not divisible by 4.

5: Last Digit is 0 or 5

A number is divisible by 5 if its last digit is either 0 or 5.

• Example: 120 is divisible by 5 (last digit is 0), while 345 is also divisible by 5 (last digit is 5). 237 is not divisible by 5 because its last digit is 7.

6: Divisible by 2 and 3

A number is divisible by 6 if it is divisible by both 2 and 3.

• Example: The number 126 is divisible by 2 (because its last digit is 6) and divisible by 3 (because 1+2+6=9, and 9 is divisible by 3). Therefore, 126 is divisible by 6.

8: Last Three Digits

A number is divisible by 8 if its last three digits form a number divisible by 8.

• Example: Consider 1328. The last three digits, 328, are divisible by 8 (328/8 = 41). Therefore, 1328 is divisible by 8. However, 2465 is not divisible by 8 because 465 is not divisible by 8.

9: Sum of Digits

A number is divisible by 9 if the sum of its digits is divisible by 9.

• Example: Consider 1359. The sum of its digits is 1+3+5+9 = 18. Since 18 is divisible by 9, 1359 is divisible by 9.

10: Last Digit is 0

A number is divisible by 10 if its last digit is 0.

• Example: 120, 560, 1000 are all divisible by 10.

11: Alternating Sum of Digits

A number is divisible by 11 if the alternating sum of its digits is divisible by 11. This means you add the digits in odd positions, subtract the digits in even positions, and check if the result is divisible by 11.

• Example: Let's examine 121. 1 - 2 + 1 = 0. Since 0 is divisible by 11, 121 is divisible by 11. Now let's look at 242: 2 - 4 + 2 = 0. Again, divisible by 11!

Divisibility Rules Worksheet (Create your own PDF from this!)

This section provides questions to help you practice using the divisibility rules. You can use this as a template to create your own PDF worksheet.

Instructions: Determine whether the following numbers are divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 11. Circle all applicable divisors for each number.

Number	Divisible by:
1260	2, 3, 4, 5, 6, 8, 9, 10, 11
4536	2, 3, 4, 6, 9
7855	5, 11
9113	3, 11
123456	2, 3, 4, 6, 8, 9, 12
27720	2, 3, 4, 5, 6, 8, 10
389500	2, 3, 5, 10
1001	7, 11, 13
54480	2, 3, 4, 5, 6, 8, 10
73331	11

Note: Many numbers will be divisible by more than one number. For example, a number divisible by both 2 and 3 is automatically divisible by 6.

Answers to the Divisibility Rules Worksheet

Here are the answers to the worksheet above. Remember to check your work using long division to confirm the results. This reinforcement is crucial for solidifying your understanding.

Number	Divisible by:
1260	2, 3, 4, 5, 6, 9, 10
4536	2, 3, 4, 6, 9
7855	5, 11
9113	3, 11
123456	2, 3, 4, 6, 8
27720	2, 3, 4, 5, 6, 8, 10
389500	2, 5, 10
1001	7, 11, 13
54480	2, 3, 4, 5, 6, 8, 10
73331	11

Advanced Divisibility Rules

While the rules above cover the most common divisors, there are also divisibility rules for less frequently used numbers. These often involve more complex patterns and calculations.

Divisibility Rule for 7:

This rule is less straightforward. One method involves subtracting twice the last digit from the remaining number. Repeat the process until you get a number easily divisible by 7 or recognize a pattern.

• Example: Let’s check if 91 is divisible by 7. 91 becomes 9 - (2 * 1) = 7. Since 7 is divisible by 7, 91 is divisible by 7.

Divisibility Rule for 13:

Similar to the rule for 7, this involves a subtractive process. Subtract 4 times the last digit from the rest of the number. Repeat until you reach a known multiple of 13.

• Example: Checking 104: 10 - (4*4) = 10-16=-6. We must use the absolute value. 6 is not a multiple of 13. But 104/13 =8, so 104 is divisible by 13.

These more complex rules require practice and may not always be the most efficient method. Sometimes, traditional long division might be quicker.

Why are Divisibility Rules Important?

Mastering divisibility rules offers numerous advantages:

• Faster Calculations: They significantly speed up calculations, especially when dealing with large numbers.
• Improved Problem-Solving: Understanding divisibility enhances your ability to solve various mathematical problems, including fraction simplification and factor finding.
• Stronger Number Sense: Regular use of these rules improves your overall understanding of number properties and relationships.
• Foundation for Algebra: They build a strong foundation for more advanced mathematical concepts in algebra and beyond.

Conclusion

Divisibility rules are invaluable tools for enhancing your mathematical skills. By understanding and practicing these rules, you’ll improve your calculation speed, problem-solving abilities, and overall number sense. Remember, consistent practice is key to mastering these rules. Create your own worksheets based on the example provided and keep challenging yourself with increasingly complex numbers. Good luck and happy calculating!