Divisibility Rules Worksheet | Grade1to6.com - Free Printable

The image provides a summary of Divisibility Rules for numbers from 2 to 12. These rules help determine whether a number is divisible by a given divisor without performing actual division. Below, I will explain each rule in detail:

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Divisibility Rules

#### a. Divisible by 2
- Rule: A number is divisible by 2 if its last digit is \(0, 2, 4, 6,\) or \(8\).
- Explanation: Even numbers are divisible by 2. The last digit determines whether the number is even.

#### b. Divisible by 3
- Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Explanation: Add all the digits of the number. If the resulting sum is divisible by 3, then the original number is also divisible by 3.

#### c. Divisible by 4
- Rule: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Explanation: Focus only on the last two digits of the number. If this two-digit number is divisible by 4, the entire number is divisible by 4.

#### d. Divisible by 5
- Rule: A number is divisible by 5 if its last digit is \(0\) or \(5\).
- Explanation: Numbers ending in \(0\) or \(5\) are multiples of 5.

#### e. Divisible by 6
- Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
- Explanation: Combine the rules for divisibility by 2 and 3. The number must be even (last digit is \(0, 2, 4, 6,\) or \(8\)) and the sum of its digits must be divisible by 3.

#### f. Divisible by 7
- Rule: Cross off the last digit, double it, and subtract it from the remaining number. Repeat if necessary. If the resulting number is divisible by 7, the original number is divisible by 7.
- Explanation: This is an iterative process. For example, for the number 161:
1. Cross off the last digit: \(16\), double it: \(2 \times 1 = 2\), subtract: \(16 - 2 = 14\).
2. Since 14 is divisible by 7, 161 is divisible by 7.

#### g. Divisible by 8
- Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
- Explanation: Focus only on the last three digits of the number. If this three-digit number is divisible by 8, the entire number is divisible by 8.

#### h. Divisible by 9
- Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Explanation: Similar to the rule for 3, but here the sum of the digits must be divisible by 9.

#### i. Divisible by 10
- Rule: A number is divisible by 10 if its last digit is \(0\).
- Explanation: Numbers ending in \(0\) are multiples of 10.

#### j. Divisible by 11
- Rule: Subtract the last digit from the number formed by the remaining digits. If the resulting number is divisible by 11, the original number is divisible by 11.
- Explanation: For example, for the number 121:
1. Separate the last digit: \(12 - 1 = 11\).
2. Since 11 is divisible by 11, 121 is divisible by 11.

#### k. Divisible by 12
- Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
- Explanation: Combine the rules for divisibility by 3 and 4. The sum of the digits must be divisible by 3, and the last two digits must form a number divisible by 4.

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Example Application

Let's apply these rules to the number 1,232:

1. Divisible by 2: Last digit is \(2\) (even). Yes.
2. Divisible by 3: Sum of digits \(1 + 2 + 3 + 2 = 8\). \(8\) is not divisible by 3. No.
3. Divisible by 4: Last two digits are \(32\). \(32 \div 4 = 8\). Yes.
4. Divisible by 5: Last digit is not \(0\) or \(5\). No.
5. Divisible by 6: Not divisible by 3. No.
6. Divisible by 7: Cross off last digit: \(123\), double it: \(2 \times 2 = 4\), subtract: \(123 - 4 = 119\). Repeat: \(11 - 18 = -7\). \(-7\) is divisible by 7. Yes.
7. Divisible by 8: Last three digits are \(232\). \(232 \div 8 = 29\). Yes.
8. Divisible by 9: Sum of digits \(1 + 2 + 3 + 2 = 8\). \(8\) is not divisible by 9. No.
9. Divisible by 10: Last digit is not \(0\). No.
10. Divisible by 11: Subtract last digit: \(123 - 2 = 121\). \(121 \div 11 = 11\). Yes.
11. Divisible by 12: Not divisible by 3. No.

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Final Answer

The divisibility rules are explained above. Use them to check divisibility for any number as needed.

Boxed Answer:
\[
\boxed{\text{See explanations above}}
\]