Divisibility Rules

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Created by: Team Maths - Examples.com, Last Updated: July 19, 2024

Divisibility Rules

What are Divisibility Rules?

Divisibility rules are a set of simple shortcuts that help determine whether one number is divisible by another without performing full division. These rules make it easy to quickly identify factors of large numbers, which is particularly useful in simplifying fractions, finding greatest common divisors, and solving problems involving multiples.

Divisibility Rules 2 to 10

Divisibility by 2

  • Rule: A number is divisible by 2 if its last digit (units place) is even (0, 2, 4, 6, or 8).
  • Example: 1246 is divisible by 2 because the last digit is 6.

Divisibility by 3

  • Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: 471 (4 + 7 + 1 = 12, which is divisible by 3).

Divisibility by 4

  • Rule: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Example: 1296 is divisible by 4 because the last two digits form 96, which is divisible by 4.

Divisibility by 5

  • Rule: A number is divisible by 5 if its last digit is 0 or 5.
  • Example: 785 is divisible by 5 because the last digit is 5.

Divisibility by 6

  • Rule: A number is divisible by 6 if it meets the rules for divisibility by both 2 and 3.
  • Example: 222 is divisible by 6 because it is divisible by both 2 (last digit is 2) and 3 (sum of digits is 6).

Divisibility by 7

  • Rule: To check for divisibility by 7, double the last digit, subtract it from the rest of the number, and if the result is divisible by 7 (or is 0), then so is the original number.
  • Example: For 203, double the last digit (6), subtract it from 20 to get 14, which is divisible by 7.

Divisibility by 8

  • Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
  • Example: 3128 is divisible by 8 because the last three digits, 128, are divisible by 8.

Divisibility by 9

  • Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example: 5130 (5 + 1 + 3 + 0 = 9, which is divisible by 9).

Divisibility Rules Chart and Examples

DivisorRuleExample
2A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).136: Last digit is 6 (even), so 136 is divisible by 2.
3A number is divisible by 3 if the sum of its digits is divisible by 3.111: Sum of digits = 1+1+1 = 3, which is divisible by 3.
4A number is divisible by 4 if the last two digits form a number that’s divisible by 4.1224: Last two digits are 24, which is divisible by 4.
5A number is divisible by 5 if its last digit is 0 or 5.145: Last digit is 5, so 145 is divisible by 5.
6A number is divisible by 6 if it is divisible by both 2 and 3.132: Divisible by 2 (last digit 2 is even) and by 3 (sum of digits 1+3+2 = 6).
7To check divisibility by 7, double the last digit, subtract it from the rest of the number; if the result is divisible by 7, so is the original number.343: 34 – 2*3 = 28, which is divisible by 7.
8A number is divisible by 8 if the last three digits form a number that’s divisible by 8.1600: Last three digits are 600, which is divisible by 8.
9A number is divisible by 9 if the sum of its digits is divisible by 9.2349: Sum of digits = 2+3+4+9 = 18, which is divisible by 9.
10A number is divisible by 10 if its last digit is 0.230: Last digit is 0, so 230 is divisible by 10.

Divisibility by 10

  • Rule: A number is divisible by 10 if its last digit is 0.
  • Example: 430 is divisible by 10 because the last digit is 0.

Divisibility Rules for Prime Numbers

Divisibility by 2

  • Explanation: A number is divisible by 2 if its last digit is even, which includes 0, 2, 4, 6, or 8. This rule is one of the simplest to apply since you only need to observe the final digit of the number.
  • Example: The number 148 is divisible by 2 because its last digit is 8.

Divisibility by 3

  • Explanation: A number is divisible by 3 if the sum of its digits is divisible by 3. This requires adding up all the digits of the number and checking the sum’s divisibility.
  • Example: For the number 123, the sum of the digits is 6 (1 + 2 + 3), which is divisible by 3.

Divisibility by 5

  • Explanation: A number is divisible by 5 if its last digit is 0 or 5. This rule is also straightforward and requires only a quick glance at the number’s last digit.
  • Example: The number 320 ends in 0, indicating it is divisible by 5.

Divisibility by 7

  • Explanation: To determine divisibility by 7, take the last digit, double it, and subtract this value from the rest of the number. If the result is divisible by 7 (or equals zero), then the original number is also divisible by 7.
  • Example: Consider the number 161; double the last digit (1 x 2 = 2) and subtract from the remaining number (16 – 2 = 14). Since 14 is divisible by 7, so is 161

Divisibility Rules of 13, 17, and 19 Examples

Divisibility by 13

  • Explanation: To check if a number is divisible by 13, multiply the last digit by 4 and add this result to the rest of the number. If the resultant number is divisible by 13, then the original number is as well.
  • Example: Consider the number 247. Multiply the last digit (7) by 4 to get 28, add this to 24 (the remaining part of the number), which gives 52. Since 52 is divisible by 13 (52 ÷ 13 = 4), 247 is also divisible by 13.

Divisibility by 17

  • Explanation: To determine divisibility by 17, multiply the last digit by 5 and subtract this from the rest of the number. If the resulting number is divisible by 17, then the original number is too.
  • Example: Take the number 629. Multiply the last digit (9) by 5 to get 45, subtract this from 62 (the remaining part of the number), resulting in 17. Since 17 is divisible by 17 (17 ÷ 17 = 1), 629 is divisible by 17.

Divisibility by 19

  • Explanation: For divisibility by 19, multiply the last digit by 2 and subtract it from the rest of the number. If the result is divisible by 19, then so is the original number.
  • Example: Consider the number 437. Multiply the last digit (7) by 2 to get 14, subtract this from 43 (the remaining part of the number), which gives 29. Since 29 is not divisible by 19, 437 is not divisible by 19 either.

30+ Numbers divisibility Examples

Below is a list of 30 examples demonstrating the divisibility rules for various numbers. These examples are designed to help educators and students easily understand how to apply these rules in practice.

  1. 24 is divisible by 2 because the last digit (4) is even.
  2. 35 is divisible by 5 because the last digit is 5.
  3. 144 is divisible by 3 because the sum of its digits (1+4+4) equals 9, which is divisible by 3.
  4. 128 is divisible by 4 because the last two digits (28) form a number that is divisible by 4.
  5. 210 is divisible by 6 because it is divisible by both 2 (last digit 0 is even) and 3 (sum of digits 2+1+0 = 3).
  6. 182 is divisible by 7 because 18 – 2×2 = 14, which is divisible by 7.
  7. 256 is divisible by 8 because the last three digits (256) form a number that is divisible by 8.
  8. 369 is divisible by 9 because the sum of its digits (3+6+9 = 18) is divisible by 9.
  9. 140 is divisible by 10 because its last digit is 0.
  10. 462 is divisible by 2 because the last digit (2) is even.
  11. 975 is divisible by 5 because the last digit is 5.
  12. 111 is divisible by 3 because the sum of its digits (1+1+1) equals 3, which is divisible by 3.
  13. 112 is divisible by 4 because the last two digits (12) form a number that is divisible by 4.
  14. 312 is divisible by 6 because it is divisible by both 2 (last digit 2 is even) and 3 (sum of digits 3+1+2 = 6).
  15. 203 is divisible by 7 because 20 – 2×3 = 14, which is divisible by 7.
  16. 1200 is divisible by 8 because the last three digits (200) form a number that is divisible by 8.
  17. 558 is divisible by 9 because the sum of its digits (5+5+8 = 18) is divisible by 9.
  18. 230 is divisible by 10 because its last digit is 0.
  19. 198 is divisible by 2 because the last digit (8) is even.
  20. 625 is divisible by 5 because the last digit is 5.
  21. 276 is divisible by 3 because the sum of its digits (2+7+6) equals 15, which is divisible by 3.
  22. 808 is divisible by 4 because the last two digits (08) form a number that is divisible by 4.
  23. 534 is divisible by 6 because it is divisible by both 2 (last digit 4 is even) and 3 (sum of digits 5+3+4 = 12).
  24. 637 is divisible by 7 because 63 – 2×7 = 49, which is divisible by 7.
  25. 1040 is divisible by 8 because the last three digits (040) form a number that is divisible by 8.
  26. 459 is divisible by 9 because the sum of its digits (4+5+9 = 18) is divisible by 9.
  27. 850 is divisible by 10 because its last digit is 0.
  28. 596 is divisible by 2 because the last digit (6) is even.
  29. 940 is divisible by 5 because the last digit is 0.
  30. 981 is divisible by 3 because the sum of its digits (9+8+1) equals 18, which is divisible by 3.

Divisibility Rules Tips and Tricks:

Mastering divisibility rules not only helps with simplifying mathematical operations but also enhances speed and accuracy in problem-solving. Here are some practical tips and tricks to effectively use divisibility rules:

Understand the Concept

  • Foundation: Grasp the basic idea behind each divisibility rule. Knowing why these rules work can help you remember them better and apply them more confidently.

Practice Regularly

  • Routine: Regular practice with different numbers helps solidify your understanding and speed up the process of checking divisibility.

Use Summation for Multiple Digits

  • Digits Addition: For divisibility rules involving sums of digits (like those for 3 and 9), practice quick mental addition. The faster you can sum digits, the quicker you can apply these rules.

Memorize Key Multiples

  • Shortcut: For divisibility by numbers such as 7, 11, 13, or 17, knowing a few key multiples can help you quickly determine if a larger number is divisible by these primes without doing all the calculations.

Combine Rules for Efficiency

  • Layered Checks: Use multiple divisibility rules in combination when checking divisibility by composite numbers. For example, a number divisible by 6 must pass the rules for both 2 and 3.

Check from Simple to Complex

  • Strategy: Start with the simplest divisibility rules (like 2 and 5) before moving to more complex ones. This can often help eliminate possibilities early on.

Utilize the Subtraction Trick

  • Helpful for Large Numbers: For divisibility by 7, 11, 13, or 17, use their respective tricks involving subtraction or addition of multiplied last digits. These can simplify large numbers quickly.

Look for Patterns

  • Pattern Recognition: As you practice, you’ll start to see patterns in numbers that make it easier to apply divisibility rules. For instance, you might begin to quickly recognize when a number is likely not divisible by a particular prime based on its digit configuration.

Employ Visual Aids

  • Visual Learning: Use charts or tables that list key multiples or summarize rules until they are memorized. Visual aids can be particularly helpful in classrooms or study sessions.

Teach Others

  • Reinforcement: Teaching divisibility rules to others is a great way to deepen your own understanding and discover any gaps in your knowledge.

Practice Examples

Example 1: Check if 154 is divisible by 2

Solution:

Given, 154 is a number.

If the last digit of 154 is divisible by 2, then 154 is also divisible by 2.

The last digit of 154 is 4, which is divisible by 2, such that;

4/2 = 2

Hence, 154 satisfies the divisibility rule for 2.

Example 2: Check if 629 is divisible by 4 or not

Solution:

As we can see, the last two digits of 629 are 29, which is not divisible by 4.

Hence, 629 is not divisible by 4.

Example 3: Check if 276 is divisible by 3

Solution:

Given, 276 is a number.

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

The sum of the digits in 276 is 2 + 7 + 6 = 15, which is divisible by 3, such that;

15/3 = 5

Hence, 276 satisfies the divisibility rule for 3.

Example 4: Check if 481 is divisible by 5

Solution:

To determine if 481 is divisible by 5, we check the last digit of the number.

The last digit of 481 is 1, which is not divisible by 5.

Hence, 481 is not divisible by 5.

FAQs

What are the Basic Rules of Divisibility?

The basic rules of divisibility help determine if one number is divisible by another without performing division. They include rules for 2, 3, 4, 5, 6, 9, and 10, such as a number being divisible by 2 if it ends in an even number.

What are the 12 Divisibility Rules?

The 12 divisibility rules include criteria for 1 through 10, plus 12 and 15. They describe shortcuts to check divisibility, such as a number ending in 0 being divisible by 10, and a number whose digit sum is divisible by 3 being divisible by 3.

What is the 7 Rule Divisibility?

The rule for divisibility by 7 involves taking the last digit of the number, doubling it, subtracting this from the rest of the number, and if the result is divisible by 7 (or is zero), then the original number is too.

Is 33 Divisible by 3?

Yes, 33 is divisible by 3. The sum of its digits (3 + 3) is 6, which is divisible by 3.

Is 10824 Divisible by 11?

Yes, 10824 is divisible by 11. The alternating sum of its digits (1 – 0 + 8 – 2 + 4 = 11) is divisible by 11.

What are Divisibility Rules for Kids?

Divisibility rules for kids provide easy ways to determine if one number can be evenly divided by another using simple checks, like a number ending in 0 or 5 being divisible by 5, or summing digits to check for divisibility by 3.

What is the Easiest Divisibility Rule?

The easiest divisibility rule is for 10; a number is divisible by 10 if it ends in 0. This rule is straightforward and requires only a quick glance at the number’s last digit.

What is the Easiest Way to Remember Divisibility Rules?

The easiest way to remember divisibility rules is by associating each rule with a simple characteristic of the number, such as its last digit or the sum of its digits, and using mnemonic devices or rhymes.

How to Check Divisibility?

To check divisibility, apply the specific rule for the divisor: check the last digit(s), sum the digits, or perform simple arithmetic manipulations as specified by the divisibility rules for the number in question.

Is Zero Divisible by 2?

Yes, zero is divisible by 2. In mathematics, zero divided by any non-zero integer is zero, which satisfies the condition of leaving no remainder.

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