Divisibility rules 4, 8, 11 and 12 worksheet - Free Printable

To solve this problem, we need to use the divisibility rules for the numbers 4, 8, 12, and 11. Let's go through each number step by step.

Divisibility Rules:

1. Divisible by 4: A number is divisible by 4 if the last two digits form a number that is divisible by 4.
2. Divisible by 8: A number is divisible by 8 if the last three digits form a number that is divisible by 8.
3. Divisible by 12: A number is divisible by 12 if it is divisible by both 3 and 4.
- Divisible by 3: The sum of the digits is divisible by 3.
- Divisible by 4: The last two digits form a number divisible by 4.
4. Divisible by 11: A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11 (including 0).

Given Numbers: 3896, 4672, 7656, 8888, 7326

#### Step-by-Step Analysis:

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Number: 3896

1. Divisible by 4: Last two digits are 96. \( 96 \div 4 = 24 \) (YES)
2. Divisible by 8: Last three digits are 896. \( 896 \div 8 = 112 \) (YES)
3. Divisible by 12:
- Sum of digits: \( 3 + 8 + 9 + 6 = 26 \). Not divisible by 3 (NO).
- Since it is not divisible by 3, it is not divisible by 12 (NO).
4. Divisible by 11:
- Odd-position digits: 3, 9 → Sum = 3 + 9 = 12
- Even-position digits: 8, 6 → Sum = 8 + 6 = 14
- Difference: \( 12 - 14 = -2 \). Not divisible by 11 (NO).

Result for 3896: YES, YES, NO, NO

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Number: 4672

1. Divisible by 4: Last two digits are 72. \( 72 \div 4 = 18 \) (YES)
2. Divisible by 8: Last three digits are 672. \( 672 \div 8 = 84 \) (YES)
3. Divisible by 12:
- Sum of digits: \( 4 + 6 + 7 + 2 = 19 \). Not divisible by 3 (NO).
- Since it is not divisible by 3, it is not divisible by 12 (NO).
4. Divisible by 11:
- Odd-position digits: 4, 7 → Sum = 4 + 7 = 11
- Even-position digits: 6, 2 → Sum = 6 + 2 = 8
- Difference: \( 11 - 8 = 3 \). Not divisible by 11 (NO).

Result for 4672: YES, YES, NO, NO

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Number: 7656

1. Divisible by 4: Last two digits are 56. \( 56 \div 4 = 14 \) (YES)
2. Divisible by 8: Last three digits are 656. \( 656 \div 8 = 82 \) (YES)
3. Divisible by 12:
- Sum of digits: \( 7 + 6 + 5 + 6 = 24 \). Divisible by 3 (YES).
- Last two digits are 56, which is divisible by 4 (YES).
- Since it is divisible by both 3 and 4, it is divisible by 12 (YES).
4. Divisible by 11:
- Odd-position digits: 7, 5 → Sum = 7 + 5 = 12
- Even-position digits: 6, 6 → Sum = 6 + 6 = 12
- Difference: \( 12 - 12 = 0 \). Divisible by 11 (YES).

Result for 7656: YES, YES, YES, YES

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Number: 8888

1. Divisible by 4: Last two digits are 88. \( 88 \div 4 = 22 \) (YES)
2. Divisible by 8: Last three digits are 888. \( 888 \div 8 = 111 \) (YES)
3. Divisible by 12:
- Sum of digits: \( 8 + 8 + 8 + 8 = 32 \). Not divisible by 3 (NO).
- Since it is not divisible by 3, it is not divisible by 12 (NO).
4. Divisible by 11:
- Odd-position digits: 8, 8 → Sum = 8 + 8 = 16
- Even-position digits: 8, 8 → Sum = 8 + 8 = 16
- Difference: \( 16 - 16 = 0 \). Divisible by 11 (YES).

Result for 8888: YES, YES, NO, YES

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Number: 7326

1. Divisible by 4: Last two digits are 26. \( 26 \div 4 = 6.5 \) (NO)
2. Divisible by 8: Last three digits are 326. \( 326 \div 8 = 40.75 \) (NO)
3. Divisible by 12:
- Sum of digits: \( 7 + 3 + 2 + 6 = 18 \). Divisible by 3 (YES).
- Last two digits are 26, which is not divisible by 4 (NO).
- Since it is not divisible by 4, it is not divisible by 12 (NO).
4. Divisible by 11:
- Odd-position digits: 7, 2 → Sum = 7 + 2 = 9
- Even-position digits: 3, 6 → Sum = 3 + 6 = 9
- Difference: \( 9 - 9 = 0 \). Divisible by 11 (YES).

Result for 7326: NO, NO, NO, YES

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Final Answer:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{GIVEN} & 4 & 8 & 12 & 11 \\
\hline
3896 & \text{YES} & \text{YES} & \text{NO} & \text{NO} \\
4672 & \text{YES} & \text{YES} & \text{NO} & \text{NO} \\
7656 & \text{YES} & \text{YES} & \text{YES} & \text{YES} \\
8888 & \text{YES} & \text{YES} & \text{NO} & \text{YES} \\
7326 & \text{NO} & \text{NO} & \text{NO} & \text{YES} \\
\hline
\end{array}
\]

\boxed{
\begin{array}{|c|c|c|c|c|}
\hline
\text{GIVEN} & 4 & 8 & 12 & 11 \\
\hline
3896 & \text{YES} & \text{YES} & \text{NO} & \text{NO} \\
4672 & \text{YES} & \text{YES} & \text{NO} & \text{NO} \\
7656 & \text{YES} & \text{YES} & \text{YES} & \text{YES} \\
8888 & \text{YES} & \text{YES} & \text{NO} & \text{YES} \\
7326 & \text{NO} & \text{NO} & \text{NO} & \text{YES} \\
\hline
\end{array}
}