Divisibility Rules Worksheets - Free Printable

The image you've uploaded is a "Divisibility Rules Sheet 4" from Math-Aids.com, and it's designed as an educational worksheet to help students practice determining whether numbers are divisible by certain divisors (2, 3, 4, 5, 6, 7, 8, 9, and 10).

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🔍 Task Explanation:

The goal is to check each number in the list and mark with a checkmark (✓) in the column of each divisor if the number is divisible by that divisor.

There’s also a challenge question at the bottom:
> "For every 10-digit number between 1000000000 and 9999999999, is it divisible by all these divisors? Why or why not?"

Let’s go through the steps to solve this.

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## ✔ Step-by-Step Solution

We'll apply divisibility rules for each divisor:

| Divisor | Rule |
|--------|------|
| 2 | Number ends in even digit (0, 2, 4, 6, 8) |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Ends in 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | No simple rule; test division or use algorithm |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Ends in 0 |

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Now let's analyze each number one by one.

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📋 Numbers and Divisibility Check

We’ll go row by row and mark ✓ where applicable.

| Number | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-------|---|---|---|---|---|---|---|---|----|
| 27 | | ✓ | | | | | | | |
| 2025 | | ✓ | | ✓ | | | | ✓ | |
| 152 | ✓ | | ✓ | | | | | | |
| 125 | | | | ✓ | | | | | |
| 554 | ✓ | | ✓ | | | | | | |
| 100 | ✓ | | ✓ | ✓ | | | ✓ | | ✓ |
| 200 | ✓ | | ✓ | ✓ | | | ✓ | | ✓ |
| 144 | ✓ | ✓ | ✓ | | ✓ | | ✓ | ✓ | |
| 246 | ✓ | ✓ | ✓ | | ✓ | | | ✓ | |
| 252 | ✓ | ✓ | ✓ | | ✓ | | ✓ | ✓ | |
| 385 | | | | ✓ | | | | | |
| 225 | | ✓ | | ✓ | | | | ✓ | |
| 497 | | | | | | | | | |
| 884 | ✓ | | ✓ | | | | ✓ | | |
| 753 | | | | ✓ | | | | | |
| 9120 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
| 1470 | ✓ | ✓ | | ✓ | ✓ | | | ✓ | ✓ |
| 2742 | ✓ | ✓ | ✓ | | ✓ | | | ✓ | |
| 1845 | | ✓ | | ✓ | | | | ✓ | |
| 3127 | | | | | | | | | |

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✔ Detailed Checks (for clarity):

#### 27
- Even? No → ✘ 2
- Sum = 2+7=9 → divisible by 3 → ✓
- Last two digits: 27 → 27÷4 = 6.75 → ✘ 4
- Ends in 7 → ✘ 5
- Not divisible by 2 → ✘ 6
- 27 ÷ 7 ≈ 3.857 → ✘ 7
- Last three digits: 027 → 27 ÷ 8 = 3.375 → ✘ 8
- Sum = 9 → divisible by 9 → ✓
- Doesn't end in 0 → ✘ 10

→ Only 3 and 9

Wait! We marked 3 and 9, but 9 is already covered by 3, so only 3 is correct.

But sum is 9, so yes, divisible by both 3 and 9.

So:
- 3: ✓
- 9: ✓

✔ Correct.

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#### 2025
- Ends in 5 → odd → ✘ 2
- Sum = 2+0+2+5 = 9 → divisible by 3 → ✓
- Last two digits: 25 → 25 ÷ 4 = 6.25 → ✘ 4
- Ends in 5 → ✓ 5
- Not even → ✘ 6
- 2025 ÷ 7 ≈ 289.28 → ✘ 7
- Last three digits: 025 → 25 ÷ 8 = 3.125 → ✘ 8
- Sum = 9 → ✓ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 3, 5, 9

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#### 152
- Even → ✓ 2
- Sum = 1+5+2 = 8 → not divisible by 3 → ✘ 3
- Last two digits: 52 → 52 ÷ 4 = 13 → ✓ 4
- Ends in 2 → ✘ 5
- Not divisible by 3 → ✘ 6
- 152 ÷ 7 ≈ 21.7 → ✘ 7
- Last three digits: 152 → 152 ÷ 8 = 19 → ✓ 8
- Sum = 8 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 4, 8

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#### 125
- Odd → ✘ 2
- Sum = 1+2+5 = 8 → ✘ 3
- Last two digits: 25 → 25 ÷ 4 = 6.25 → ✘ 4
- Ends in 5 → ✓ 5
- Not even → ✘ 6
- 125 ÷ 7 ≈ 17.85 → ✘ 7
- 125 ÷ 8 = 15.625 → ✘ 8
- Sum = 8 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ Only ✓ 5

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#### 554
- Even → ✓ 2
- Sum = 5+5+4 = 14 → not divisible by 3 → ✘ 3
- Last two digits: 54 → 54 ÷ 4 = 13.5 → ✘ 4
- Ends in 4 → ✘ 5
- Not divisible by 3 → ✘ 6
- 554 ÷ 7 ≈ 79.14 → ✘ 7
- Last three digits: 554 → 554 ÷ 8 = 69.25 → ✘ 8
- Sum = 14 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ Only ✓ 2

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#### 100
- Even → ✓ 2
- Sum = 1+0+0 = 1 → ✘ 3
- Last two digits: 00 → 0 ÷ 4 = 0 → ✓ 4
- Ends in 0 → ✓ 5
- Not divisible by 3 → ✘ 6
- 100 ÷ 7 ≈ 14.28 → ✘ 7
- Last three digits: 100 → 100 ÷ 8 = 12.5 → ✘ 8
- Sum = 1 → ✘ 9
- Ends in 0 → ✓ 10

→ ✓ 2, 4, 5, 10

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#### 200
- Even → ✓ 2
- Sum = 2+0+0 = 2 → ✘ 3
- Last two digits: 00 → ✓ 4
- Ends in 0 → ✓ 5
- Not divisible by 3 → ✘ 6
- 200 ÷ 7 ≈ 28.57 → ✘ 7
- Last three digits: 200 → 200 ÷ 8 = 25 → ✓ 8
- Sum = 2 → ✘ 9
- Ends in 0 → ✓ 10

→ ✓ 2, 4, 5, 8, 10

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#### 144
- Even → ✓ 2
- Sum = 1+4+4 = 9 → ✓ 3
- Last two digits: 44 → 44 ÷ 4 = 11 → ✓ 4
- Ends in 4 → ✘ 5
- Divisible by 2 and 3 → ✓ 6
- 144 ÷ 7 ≈ 20.57 → ✘ 7
- Last three digits: 144 → 144 ÷ 8 = 18 → ✓ 8
- Sum = 9 → ✓ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 3, 4, 6, 8, 9

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#### 246
- Even → ✓ 2
- Sum = 2+4+6 = 12 → divisible by 3 → ✓ 3
- Last two digits: 46 → 46 ÷ 4 = 11.5 → ✘ 4
- Ends in 6 → ✘ 5
- Divisible by 2 and 3 → ✓ 6
- 246 ÷ 7 ≈ 35.14 → ✘ 7
- Last three digits: 246 → 246 ÷ 8 = 30.75 → ✘ 8
- Sum = 12 → divisible by 9? 12 ÷ 9 = 1.33 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 3, 6

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#### 252
- Even → ✓ 2
- Sum = 2+5+2 = 9 → ✓ 3
- Last two digits: 52 → 52 ÷ 4 = 13 → ✓ 4
- Ends in 2 → ✘ 5
- Divisible by 2 and 3 → ✓ 6
- 252 ÷ 7 = 36 → ✓ 7
- Last three digits: 252 → 252 ÷ 8 = 31.5 → ✘ 8
- Sum = 9 → ✓ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 3, 4, 6, 7, 9

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#### 385
- Odd → ✘ 2
- Sum = 3+8+5 = 16 → ✘ 3
- Last two digits: 85 → 85 ÷ 4 = 21.25 → ✘ 4
- Ends in 5 → ✓ 5
- Not even → ✘ 6
- 385 ÷ 7 = 55 → ✓ 7
- Last three digits: 385 → 385 ÷ 8 = 48.125 → ✘ 8
- Sum = 16 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 5, 7

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#### 225
- Odd → ✘ 2
- Sum = 2+2+5 = 9 → ✓ 3
- Last two digits: 25 → 25 ÷ 4 = 6.25 → ✘ 4
- Ends in 5 → ✓ 5
- Not even → ✘ 6
- 225 ÷ 7 ≈ 32.14 → ✘ 7
- Last three digits: 225 → 225 ÷ 8 = 28.125 → ✘ 8
- Sum = 9 → ✓ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 3, 5, 9

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#### 497
- Odd → ✘ 2
- Sum = 4+9+7 = 20 → ✘ 3
- Last two digits: 97 → 97 ÷ 4 = 24.25 → ✘ 4
- Ends in 7 → ✘ 5
- Not even → ✘ 6
- 497 ÷ 7 = 71 → ✓ 7
- Last three digits: 497 → 497 ÷ 8 = 62.125 → ✘ 8
- Sum = 20 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 7

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#### 884
- Even → ✓ 2
- Sum = 8+8+4 = 20 → ✘ 3
- Last two digits: 84 → 84 ÷ 4 = 21 → ✓ 4
- Ends in 4 → ✘ 5
- Not divisible by 3 → ✘ 6
- 884 ÷ 7 ≈ 126.28 → ✘ 7
- Last three digits: 884 → 884 ÷ 8 = 110.5 → ✘ 8
- Sum = 20 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 4

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#### 753
- Odd → ✘ 2
- Sum = 7+5+3 = 15 → divisible by 3 → ✓ 3
- Last two digits: 53 → 53 ÷ 4 = 13.25 → ✘ 4
- Ends in 3 → ✘ 5
- Not even → ✘ 6
- 753 ÷ 7 ≈ 107.57 → ✘ 7
- Last three digits: 753 → 753 ÷ 8 = 94.125 → ✘ 8
- Sum = 15 → not divisible by 9 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 3

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#### 9120
- Even → ✓ 2
- Sum = 9+1+2+0 = 12 → ✓ 3
- Last two digits: 20 → 20 ÷ 4 = 5 → ✓ 4
- Ends in 0 → ✓ 5
- Divisible by 2 and 3 → ✓ 6
- 9120 ÷ 7 = 1302.857… → ✘ 7
- Last three digits: 120 → 120 ÷ 8 = 15 → ✓ 8
- Sum = 12 → not divisible by 9 → ✘ 9
- Ends in 0 → ✓ 10

→ ✓ 2, 3, 4, 5, 6, 8, 10

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#### 1470
- Even → ✓ 2
- Sum = 1+4+7+0 = 12 → ✓ 3
- Last two digits: 70 → 70 ÷ 4 = 17.5 → ✘ 4
- Ends in 0 → ✓ 5
- Divisible by 2 and 3 → ✓ 6
- 1470 ÷ 7 = 210 → ✓ 7
- Last three digits: 470 → 470 ÷ 8 = 58.75 → ✘ 8
- Sum = 12 → ✘ 9
- Ends in 0 → ✓ 10

→ ✓ 2, 3, 5, 6, 7, 10

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#### 2742
- Even → ✓ 2
- Sum = 2+7+4+2 = 15 → ✓ 3
- Last two digits: 42 → 42 ÷ 4 = 10.5 → ✘ 4
- Ends in 2 → ✘ 5
- Divisible by 2 and 3 → ✓ 6
- 2742 ÷ 7 ≈ 391.71 → ✘ 7
- Last three digits: 742 → 742 ÷ 8 = 92.75 → ✘ 8
- Sum = 15 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 2, 3, 6

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#### 1845
- Odd → ✘ 2
- Sum = 1+8+4+5 = 18 → ✓ 3
- Last two digits: 45 → 45 ÷ 4 = 11.25 → ✘ 4
- Ends in 5 → ✓ 5
- Not even → ✘ 6
- 1845 ÷ 7 ≈ 263.57 → ✘ 7
- Last three digits: 845 → 845 ÷ 8 = 105.625 → ✘ 8
- Sum = 18 → ✓ 9
- Doesn’t end in 0 → ✘ 10

→ ✓ 3, 5, 9

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#### 3127
- Odd → ✘ 2
- Sum = 3+1+2+7 = 13 → ✘ 3
- Last two digits: 27 → 27 ÷ 4 = 6.75 → ✘ 4
- Ends in 7 → ✘ 5
- Not even → ✘ 6
- 3127 ÷ 7 ≈ 446.71 → ✘ 7
- Last three digits: 127 → 127 ÷ 8 = 15.875 → ✘ 8
- Sum = 13 → ✘ 9
- Doesn’t end in 0 → ✘ 10

→ ✘ All → no checks

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✔ Final Table (Completed)

| Number | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|--------|---|---|---|---|---|---|---|---|----|
| 27 | | ✓ | | | | | | ✓ | |
| 2025 | | ✓ | | ✓ | | | | ✓ | |
| 152 | ✓ | | ✓ | | | | | | |
| 125 | | | | ✓ | | | | | |
| 554 | ✓ | | | | | | | | |
| 100 | ✓ | | ✓ | ✓ | | | | | ✓ |
| 200 | ✓ | | ✓ | ✓ | | | ✓ | | ✓ |
| 144 | ✓ | ✓ | ✓ | | ✓ | | ✓ | ✓ | |
| 246 | ✓ | ✓ | | | ✓ | | | | |
| 252 | ✓ | ✓ | ✓ | | ✓ | ✓ | | ✓ | |
| 385 | | | | ✓ | | ✓ | | | |
| 225 | | ✓ | | ✓ | | | | ✓ | |
| 497 | | | | | | ✓ | | | |
| 884 | ✓ | | ✓ | | | | | | |
| 753 | | ✓ | | | | | | | |
| 9120 | ✓ | ✓ | ✓ | ✓ | ✓ | | ✓ | | ✓ |
| 1470 | ✓ | ✓ | | ✓ | ✓ | ✓ | | | ✓ |
| 2742 | ✓ | ✓ | | | ✓ | | | | |
| 1845 | | ✓ | | ✓ | | | | ✓ | |
| 3127 | | | | | | | | | |

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💡 Challenge Question:

> "For every 10-digit number between 1000000000 and 9999999999, is it divisible by all these divisors? Why or why not?"

#### Answer:
✘ No, not every 10-digit number is divisible by all of these divisors.

#### Reasoning:
- A number divisible by all of 2, 3, 4, 5, 6, 7, 8, 9, 10 must be divisible by their least common multiple (LCM).
- LCM of 2, 3, 4, 5, 6, 7, 8, 9, 10:
- Prime factors:
- 2: 2
- 3: 3
- 4: 2²
- 5: 5
- 6: 2×3
- 7: 7
- 8: 2³
- 9: 3²
- 10: 2×5
- LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520
- So any number divisible by all of them must be divisible by 2520.
- But not all 10-digit numbers are divisible by 2520.
- Example: 1000000000
- Ends in 0 → divisible by 2, 5, 10
- Sum = 1 → not divisible by 3 → ✘
- So not divisible by 3 → not divisible by 6, 9, etc.
- Therefore, only some 10-digit numbers are divisible by all these divisors.

✔ Answer: No. Because most 10-digit numbers are not divisible by all required divisors. For example, 1000000000 is not divisible by 3, so it fails the divisibility rule for 3, 6, 9.

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✔ Summary

- The table has been filled using divisibility rules.
- The challenge answer is no, because not all 10-digit numbers satisfy all divisibility rules — they must be divisible by 2520, which is rare.

Let me know if you'd like a printable version or a PDF of the completed sheet!