Divisibility Rules Worksheet: A Comprehensive Guide for Students

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Are you struggling with understanding divisibility rules? Do you find it challenging to determine if a number is divisible by another number without performing lengthy calculations? If so, you're in the right place! In this article, we will explore divisibility rules in a comprehensive manner, providing you with a clear understanding of how they work. Whether you're a student preparing for a math exam or someone who wants to brush up on their math skills, this article will serve as an invaluable resource. So let's dive in!

Table of Contents

1. Introduction to Divisibility Rules
2. Divisibility Rule for 2
3. Divisibility Rule for 3
4. Divisibility Rule for 4
5. Divisibility Rule for 5
6. Divisibility Rule for 6
7. Divisibility Rule for 7
8. Divisibility Rule for 8
9. Divisibility Rule for 9
10. Divisibility Rule for 10
11. Divisibility Rule for 11
12. Divisibility Rule for 12
13. Divisibility Rule for 13
14. Divisibility Rule for 14
15. Divisibility Rule for 15

1. Introduction to Divisibility Rules

Divisibility rules are mathematical shortcuts that allow us to determine if one number is divisible by another without actually performing the division. These rules are based on specific patterns and properties of numbers. By applying these rules, we can quickly identify whether a number is divisible by another, which can be extremely helpful in various mathematical calculations and problem-solving scenarios.

2. Divisibility Rule for 2

The divisibility rule for 2 states that a number is divisible by 2 if its last digit is even, i.e., 0, 2, 4, 6, or 8. For example, the number 246 is divisible by 2 since its last digit is 6, which is an even number.

3. Divisibility Rule for 3

The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. For instance, let's consider the number 369. The sum of its digits (3 + 6 + 9) equals 18, which is divisible by 3, indicating that 369 is divisible by 3.

4. Divisibility Rule for 4

The divisibility rule for 4 states that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, the number 2,548 is divisible by 4 because the last two digits, 48, form a number divisible by 4.

5. Divisibility Rule for 5

The divisibility rule for 5 states that a number is divisible by 5 if its last digit is either 0 or 5. For instance, the number 725 is divisible by 5 because its last digit is 5.

6. Divisibility Rule for 6

The divisibility rule for 6 combines the rules for divisibility by both 2 and 3. According to this rule, a number is divisible by 6 if it is divisible by both 2 and 3 simultaneously. For example, the number 594 is divisible by 6 because it is divisible by 2 and 3.

7. Divisibility Rule for 7

The divisibility rule for 7 is slightly more complex. To determine if a number is divisible by 7, double the last digit and subtract it from the number formed by the remaining digits. If the resulting number is divisible by 7, then the original number is divisible by 7 as well. Let's consider an example: 413. Double the last digit (3) to get 6. Subtracting 6 from the number formed by the remaining digits (41) gives us 41 - 6 = 35. Since 35 is divisible by 7, we can conclude that 413 is divisible by 7.

8. Divisibility Rule for 8

The divisibility rule for 8 states that a number is divisible by 8 if the number formed by its last three digits is divisible by 8. For example, let's consider the number 5,376. The last three digits, 376, form a number divisible by 8, indicating that 5,376 is divisible by 8.

9. Divisibility Rule for 9

Similar to the divisibility rule for 3, the divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9. For instance, consider the number 783. The sum of its digits (7 + 8 + 3) equals 18, which is divisible by 9, implying that 783 is divisible by 9.

10. Divisibility Rule for 10

The divisibility rule for 10 is simple. A number is divisible by 10 if it ends with a 0. For example, the number 520 is divisible by 10 since its last digit is 0.

11. Divisibility Rule for 11

The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of the digits at even positions and the sum of the digits at odd positions is either 0 or divisible by 11. Let's take the number 2,453 as an example. The sum of the digits at even positions (4 + 3) is 7, and the sum of the digits at odd positions (2 + 5) is 7 as well. Since the difference is 0, we can conclude that 2,453 is divisible by 11.

12. Divisibility Rule for 12

The divisibility rule for 12 combines the rules for divisibility by both 3 and 4. According to this rule, a number is divisible by 12 if it is divisible by both 3 and 4 simultaneously.

13. Divisibility Rule for 13

The divisibility rule for 13 is a bit more complicated. To determine if a number is divisible by 13, multiply the last digit by 4, and subtract it from the number formed by the remaining digits. If the resulting number is divisible by 13, then the original number is divisible by 13 as well.

14. Divisibility Rule for 14

The divisibility rule for 14 combines the rules for divisibility by both 2 and 7. According to this rule, a number is divisible by 14 if it is divisible by both 2 and 7 simultaneously.

15. Divisibility Rule for 15

The divisibility rule for 15 combines the rules for divisibility by both 3 and 5. According to this rule, a number is divisible by 15 if it is divisible by both 3 and 5 simultaneously.

Conclusion

Divisibility rules are valuable tools that enable us to determine if a number is divisible by another number quickly. By applying these rules, we can save time and effort in various mathematical calculations. Remember to practice these rules regularly to enhance your mathematical skills and problem-solving abilities.

FAQs (Frequently Asked Questions)

1. Why are divisibility rules important in mathematics? Divisibility rules provide quick methods to determine if a number is divisible by another number, saving time in calculations and problem-solving.

2. Can I apply multiple divisibility rules to a single number? Yes, depending on the divisibility rules, you can apply multiple rules to determine divisibility.

3. Are there divisibility rules for numbers other than those mentioned in this article? Yes, divisibility rules exist for other numbers as well, but they may not be as commonly used or straightforward as the ones discussed here.

4. How can I remember all the divisibility rules? Practice and repetition are key to remembering the divisibility rules. Regularly applying them in mathematical exercises will help you internalize the rules.

5. Are there any shortcuts for larger numbers? While the divisibility rules covered in this article are applicable to many numbers, for larger or more complex numbers, the rules may become more intricate or require additional mathematical concepts.