Divisibility Tests Worksheet :: A Transum Worksheet - Free Printable

To solve the problem, we need to determine whether each number in the left column is divisible by the numbers in the top row (2, 3, 4, 5, 6, 8, 9). We will mark a tick (√) if the number is divisible and a cross (×) if it is not. Let's go through each number step by step.

Divisibility Rules:

1. Divisible by 2: The number must be even.
2. Divisible by 3: The sum of the digits must be divisible by 3.
3. Divisible by 4: The last two digits must form a number divisible by 4.
4. Divisible by 5: The number must end in 0 or 5.
5. Divisible by 6: The number must be divisible by both 2 and 3.
6. Divisible by 8: The last three digits must form a number divisible by 8.
7. Divisible by 9: The sum of the digits must be divisible by 9.

Step-by-Step Analysis:

#### 1. 12
- 2: Even → √
- 3: Sum of digits = 1 + 2 = 3 (divisible by 3) → √
- 4: Last two digits = 12 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Divisible by both 2 and 3 → √
- 8: Last three digits = 012 (not divisible by 8) → ×
- 9: Sum of digits = 1 + 2 = 3 (not divisible by 9) → ×

#### 2. 24
- 2: Even → √
- 3: Sum of digits = 2 + 4 = 6 (divisible by 3) → √
- 4: Last two digits = 24 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Divisible by both 2 and 3 → √
- 8: Last three digits = 024 (divisible by 8) → √
- 9: Sum of digits = 2 + 4 = 6 (not divisible by 9) → ×

#### 3. 31
- 2: Odd → ×
- 3: Sum of digits = 3 + 1 = 4 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 2 or 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 3 + 1 = 4 (not divisible by 9) → ×

#### 4. 52
- 2: Even → √
- 3: Sum of digits = 5 + 2 = 7 (not divisible by 3) → ×
- 4: Last two digits = 52 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 3 → ×
- 8: Last three digits = 052 (not divisible by 8) → ×
- 9: Sum of digits = 5 + 2 = 7 (not divisible by 9) → ×

#### 5. 68
- 2: Even → √
- 3: Sum of digits = 6 + 8 = 14 (not divisible by 3) → ×
- 4: Last two digits = 68 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 3 → ×
- 8: Last three digits = 068 (divisible by 8) → √
- 9: Sum of digits = 6 + 8 = 14 (not divisible by 9) → ×

#### 6. 100
- 2: Even → √
- 3: Sum of digits = 1 + 0 + 0 = 1 (not divisible by 3) → ×
- 4: Last two digits = 00 (divisible by 4) → √
- 5: Ends in 0 → √
- 6: Not divisible by 3 → ×
- 8: Last three digits = 100 (not divisible by 8) → ×
- 9: Sum of digits = 1 + 0 + 0 = 1 (not divisible by 9) → ×

#### 7. 155
- 2: Odd → ×
- 3: Sum of digits = 1 + 5 + 5 = 11 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Ends in 5 → √
- 6: Not divisible by 2 or 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 1 + 5 + 5 = 11 (not divisible by 9) → ×

#### 8. 200
- 2: Even → √
- 3: Sum of digits = 2 + 0 + 0 = 2 (not divisible by 3) → ×
- 4: Last two digits = 00 (divisible by 4) → √
- 5: Ends in 0 → √
- 6: Not divisible by 3 → ×
- 8: Last three digits = 200 (divisible by 8) → √
- 9: Sum of digits = 2 + 0 + 0 = 2 (not divisible by 9) → ×

#### 9. 250
- 2: Even → √
- 3: Sum of digits = 2 + 5 + 0 = 7 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Ends in 0 → √
- 6: Not divisible by 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 2 + 5 + 0 = 7 (not divisible by 9) → ×

#### 10. 301
- 2: Odd → ×
- 3: Sum of digits = 3 + 0 + 1 = 4 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 2 or 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 3 + 0 + 1 = 4 (not divisible by 9) → ×

#### 11. 368
- 2: Even → √
- 3: Sum of digits = 3 + 6 + 8 = 17 (not divisible by 3) → ×
- 4: Last two digits = 68 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 3 → ×
- 8: Last three digits = 368 (divisible by 8) → √
- 9: Sum of digits = 3 + 6 + 8 = 17 (not divisible by 9) → ×

#### 12. 725
- 2: Odd → ×
- 3: Sum of digits = 7 + 2 + 5 = 14 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Ends in 5 → √
- 6: Not divisible by 2 or 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 7 + 2 + 5 = 14 (not divisible by 9) → ×

#### 13. 2456
- 2: Even → √
- 3: Sum of digits = 2 + 4 + 5 + 6 = 17 (not divisible by 3) → ×
- 4: Last two digits = 56 (divisible by 4) → √
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 3 → ×
- 8: Last three digits = 456 (divisible by 8) → √
- 9: Sum of digits = 2 + 4 + 5 + 6 = 17 (not divisible by 9) → ×

#### 14. 7281
- 2: Odd → ×
- 3: Sum of digits = 7 + 2 + 8 + 1 = 18 (divisible by 3) → √
- 4: Not divisible by 4 → ×
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 2 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 7 + 2 + 8 + 1 = 18 (divisible by 9) → √

#### 15. 284657
- 2: Odd → ×
- 3: Sum of digits = 2 + 8 + 4 + 6 + 5 + 7 = 32 (not divisible by 3) → ×
- 4: Not divisible by 4 → ×
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 2 or 3 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 2 + 8 + 4 + 6 + 5 + 7 = 32 (not divisible by 9) → ×

#### 16. 123123
- 2: Odd → ×
- 3: Sum of digits = 1 + 2 + 3 + 1 + 2 + 3 = 12 (divisible by 3) → √
- 4: Not divisible by 4 → ×
- 5: Does not end in 0 or 5 → ×
- 6: Not divisible by 2 → ×
- 8: Not divisible by 8 → ×
- 9: Sum of digits = 1 + 2 + 3 + 1 + 2 + 3 = 12 (not divisible by 9) → ×

Final Answer:

\[
\begin{array}{c|ccccccc}
& 2 & 3 & 4 & 5 & 6 & 8 & 9 \\
\hline
12 & \checkmark & \checkmark & \checkmark & \times & \checkmark & \times & \times \\
24 & \checkmark & \checkmark & \checkmark & \times & \checkmark & \checkmark & \times \\
31 & \times & \times & \times & \times & \times & \times & \times \\
52 & \checkmark & \times & \checkmark & \times & \times & \times & \times \\
68 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
100 & \checkmark & \times & \checkmark & \checkmark & \times & \times & \times \\
155 & \times & \times & \times & \checkmark & \times & \times & \times \\
200 & \checkmark & \times & \checkmark & \checkmark & \times & \checkmark & \times \\
250 & \checkmark & \times & \times & \checkmark & \times & \times & \times \\
301 & \times & \times & \times & \times & \times & \times & \times \\
368 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
725 & \times & \times & \times & \checkmark & \times & \times & \times \\
2456 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
7281 & \times & \checkmark & \times & \times & \times & \times & \checkmark \\
284657 & \times & \times & \times & \times & \times & \times & \times \\
123123 & \times & \checkmark & \times & \times & \times & \times & \times \\
\end{array}
\]

\boxed{
\begin{array}{c|ccccccc}
& 2 & 3 & 4 & 5 & 6 & 8 & 9 \\
\hline
12 & \checkmark & \checkmark & \checkmark & \times & \checkmark & \times & \times \\
24 & \checkmark & \checkmark & \checkmark & \times & \checkmark & \checkmark & \times \\
31 & \times & \times & \times & \times & \times & \times & \times \\
52 & \checkmark & \times & \checkmark & \times & \times & \times & \times \\
68 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
100 & \checkmark & \times & \checkmark & \checkmark & \times & \times & \times \\
155 & \times & \times & \times & \checkmark & \times & \times & \times \\
200 & \checkmark & \times & \checkmark & \checkmark & \times & \checkmark & \times \\
250 & \checkmark & \times & \times & \checkmark & \times & \times & \times \\
301 & \times & \times & \times & \times & \times & \times & \times \\
368 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
725 & \times & \times & \times & \checkmark & \times & \times & \times \\
2456 & \checkmark & \times & \checkmark & \times & \times & \checkmark & \times \\
7281 & \times & \checkmark & \times & \times & \times & \times & \checkmark \\
284657 & \times & \times & \times & \times & \times & \times & \times \\
123123 & \times & \checkmark & \times & \times & \times & \times & \times \\
\end{array}
}