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
- 2+7=9 → divisible by 3 → ✓
- Not even → no for 2,4,6,8
- Doesn't end in 0/5 → no for 5,10
- Not divisible by 7,9

→ Only ✓ in 3

#### 2025
- Ends in 5 → divisible by 5 → ✓
- 2+0+2+5=9 → divisible by 3,9 → ✓
- Not even → no for 2,4,6,8,10
- Not divisible by 7?

Check: 2025 ÷ 7 ≈ 289.28 → not integer → no

→ ✓ in 3,5,9

#### 152
- Even → ✓ for 2
- Last two digits: 52 → 52÷4=13 → ✓ for 4
- Not divisible by 3: 1+5+2=8 → no
- Not ending in 0/5 → no for 5,10
- 152 ÷ 7 = 21.71… → no
- 152 ÷ 8 = 19 → yes → ✓ for 8

→ ✓ in 2,4,8

#### 125
- Ends in 5 → ✓ for 5
- Odd → no for 2,4,6,8,10
- 1+2+5=8 → not divisible by 3,9
- 125 ÷ 7 ≈ 17.85 → no

→ Only ✓ in 5

#### 554
- Even → ✓ for 2
- Last two digits: 54 → 54÷4=13.5 → no
- 5+5+4=14 → not divisible by 3 → no
- Not ending in 0/5 → no for 5,10
- 554 ÷ 7 ≈ 79.14 → no
- 554 ÷ 8 = 69.25 → no

→ Only ✓ in 2

#### 100
- Even → ✓ for 2
- Ends in 0 → ✓ for 5,10
- Last two digits: 00 → divisible by 4 → ✓
- 1+0+0=1 → not divisible by 3 → no
- So not divisible by 6,9
- 100 ÷ 7 ≈ 14.28 → no
- 100 ÷ 8 = 12.5 → no

→ ✓ in 2,4,5,10

#### 200
- Even → ✓ for 2
- Ends in 0 → ✓ for 5,10
- Last two digits: 00 → divisible by 4 → ✓
- 2+0+0=2 → not divisible by 3 → no
- So not divisible by 6,9
- 200 ÷ 7 ≈ 28.57 → no
- 200 ÷ 8 = 25 → yes → ✓ for 8

→ ✓ in 2,4,5,8,10

#### 144
- Even → ✓ for 2
- 1+4+4=9 → divisible by 3,9 → ✓
- Last two digits: 44 → 44÷4=11 → ✓
- Not ending in 0/5 → no for 5,10
- Divisible by 6? Yes (divisible by 2 and 3) → ✓
- 144 ÷ 7 ≈ 20.57 → no
- 144 ÷ 8 = 18 → yes → ✓

→ ✓ in 2,3,4,6,8,9

#### 246
- Even → ✓ for 2
- 2+4+6=12 → divisible by 3 → ✓
- Last two digits: 46 → 46÷4=11.5 → no
- Not ending in 0/5 → no for 5,10
- Divisible by 6? Yes (2 and 3) → ✓
- 246 ÷ 7 ≈ 35.14 → no
- 246 ÷ 8 = 30.75 → no
- 2+4+6=12 → not divisible by 9 → no

→ ✓ in 2,3,6

#### 252
- Even → ✓ for 2
- 2+5+2=9 → divisible by 3,9 → ✓
- Last two digits: 52 → 52÷4=13 → ✓
- Not ending in 0/5 → no for 5,10
- Divisible by 6? Yes → ✓
- 252 ÷ 7 = 36 → yes → ✓
- 252 ÷ 8 = 31.5 → no

→ ✓ in 2,3,4,6,7,9

#### 385
- Odd → no for 2,4,6,8,10
- Ends in 5 → ✓ for 5
- 3+8+5=16 → not divisible by 3 → no
- 385 ÷ 7 = 55 → yes → ✓
- 385 ÷ 9 = 42.77... → no

→ ✓ in 5,7

#### 225
- Odd → no for 2,4,6,8,10
- Ends in 5 → ✓ for 5
- 2+2+5=9 → divisible by 3,9 → ✓
- 225 ÷ 7 ≈ 32.14 → no

→ ✓ in 3,5,9

#### 497
- Odd → no for 2,4,6,8,10
- 4+9+7=20 → not divisible by 3 → no
- Ends in 7 → not divisible by 5
- 497 ÷ 7 = 71 → yes → ✓
- 497 ÷ 9 ≈ 55.22 → no

→ Only ✓ in 7

#### 884
- Even → ✓ for 2
- 8+8+4=20 → not divisible by 3 → no
- Last two digits: 84 → 84÷4=21 → ✓
- Not ending in 0/5 → no for 5,10
- Not divisible by 6 (not divisible by 3)
- 884 ÷ 7 ≈ 126.28 → no
- 884 ÷ 8 = 110.5 → no

→ ✓ in 2,4

#### 753
- Odd → no for 2,4,6,8,10
- Ends in 3 → not divisible by 5
- 7+5+3=15 → divisible by 3 → ✓
- 753 ÷ 7 ≈ 107.57 → no
- 753 ÷ 9 = 83.66... → no

→ Only ✓ in 3

#### 9120
- Even → ✓ for 2
- Ends in 0 → ✓ for 5,10
- Last two digits: 20 → 20÷4=5 → ✓
- 9+1+2+0=12 → divisible by 3 → ✓
- Divisible by 6? Yes → ✓
- 9120 ÷ 7 = 1302.857... → no
- Last three digits: 120 → 120÷8=15 → ✓
- 9+1+2+0=12 → not divisible by 9 → no

→ ✓ in 2,3,4,5,6,8,10

Wait — 9120 ÷ 7 = ? Let's double-check:

7 × 1302 = 9114 → 9120 - 9114 = 6 → remainder 6 → not divisible by 7

So no for 7

But 9+1+2+0=12 → not divisible by 9 → no

→ ✓ in 2,3,4,5,6,8,10

#### 1470
- Even → ✓ for 2
- Ends in 0 → ✓ for 5,10
- Last two digits: 70 → 70÷4=17.5 → no
- 1+4+7+0=12 → divisible by 3 → ✓
- Divisible by 6? Yes → ✓
- 1470 ÷ 7 = 210 → yes → ✓
- 1470 ÷ 8 = 183.75 → no
- 1+4+7+0=12 → not divisible by 9 → no

→ ✓ in 2,3,5,6,7,10

#### 2742
- Even → ✓ for 2
- 2+7+4+2=15 → divisible by 3 → ✓
- Last two digits: 42 → 42÷4=10.5 → no
- Not ending in 0/5 → no for 5,10
- Divisible by 6? Yes → ✓
- 2742 ÷ 7 ≈ 391.7 → no
- 2742 ÷ 8 = 342.75 → no
- 2+7+4+2=15 → not divisible by 9 → no

→ ✓ in 2,3,6

#### 1845
- Odd → no for 2,4,6,8,10
- Ends in 5 → ✓ for 5
- 1+8+4+5=18 → divisible by 3,9 → ✓
- 1845 ÷ 7 ≈ 263.57 → no

→ ✓ in 3,5,9

#### 3127
- Odd → no for 2,4,6,8,10
- Ends in 7 → not divisible by 5
- 3+1+2+7=13 → not divisible by 3 → no
- 3127 ÷ 7 = 446.71 → no
- 3127 ÷ 9 ≈ 347.44 → no

→ No divisibility marks

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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.

Reasoning:

- A number divisible by all divisors (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 factorization:
- 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, only multiples of 2520 are divisible by all of them.

But most 10-digit numbers are not multiples of 2520.

For example:
- 1000000000 ÷ 2520 ≈ 396825.39 → not a whole number → not divisible
- 1000000001 → not divisible by 2 → so not divisible by all

Hence, only some 10-digit numbers are divisible by all these divisors — not all.

✔ Final Answer:
No, not every 10-digit number between 1000000000 and 9999999999 is divisible by all the given divisors. Only those divisible by the LCM of the divisors (2520) will be divisible by all of them. Most 10-digit numbers are not divisible by 2520.

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