Printable math worksheet focusing on factors and multiples, designed for educational practice.

Let's solve this step-by-step, explaining each part clearly.

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Section A

#### Question 1:
We are given a partial 100 square:

```
12 13 14 15
22 23 24 25
32 33 34 35
42 43 44 45
```

Shaded numbers: 12, 15, 24, 33, 42, 45
Circled numbers: 12, 24, 32, 44

---

##### What do the shaded numbers have in common?

List of shaded numbers: 12, 15, 24, 33, 42, 45

Let’s look at their properties:
- All end in even or odd digits — not helpful.
- Check divisibility by 3:
- 12 → 1+2=3 → divisible by 3 ✔
- 15 → 1+5=6 → divisible by 3 ✔
- 24 → 2+4=6 → divisible by 3 ✔
- 33 → 3+3=6 → divisible by 3 ✔
- 42 → 4+2=6 → divisible by 3 ✔
- 45 → 4+5=9 → divisible by 3 ✔

✔ All shaded numbers are multiples of 3.

Also, notice that they are all odd or even, but more importantly, they are all multiples of 3.

So, the shaded numbers are all multiples of 3.

---

##### What do the circled numbers have in common?

Circled numbers: 12, 24, 32, 44

Check:
- 12 → even
- 24 → even
- 32 → even
- 44 → even

All are even.

Now check if they’re multiples of something else:
- 12 = 4×3
- 24 = 4×6
- 32 = 4×8
- 44 = 4×11

Wait — 12, 24, 32, 44 — are they all divisible by 4?

- 12 ÷ 4 = 3 ✔
- 24 ÷ 4 = 6 ✔
- 32 ÷ 4 = 8 ✔
- 44 ÷ 4 = 11 ✔

Yes! All are divisible by 4.

So, the circled numbers are all multiples of 4.

But wait — let's double-check:
- 12: yes (4×3)
- 24: yes (4×6)
- 32: yes (4×8)
- 44: yes (4×11)

✔ So, they are all multiples of 4.

---

#### Question 2:

Another part of the 100 square:

```
55 56 57 58 59 60
65 66 67 68 69 70
75 76 77 78 79 80
85 86 87 88 89 90
```

Shaded numbers: 55, 60, 65, 70, 75, 80, 85, 90
Circled numbers: 55, 66, 77, 88

---

##### What do the shaded numbers have in common?

Shaded: 55, 60, 65, 70, 75, 80, 85, 90

These are all ending in 5 or 0 → so they are multiples of 5.

Check:
- 55 ÷ 5 = 11
- 60 ÷ 5 = 12
- 65 ÷ 5 = 13
- ...
- 90 ÷ 5 = 18

✔ All are multiples of 5.

So, shaded numbers are multiples of 5.

---

##### What do the circled numbers have in common?

Circled: 55, 66, 77, 88

Look at them:
- 55 = 5×11
- 66 = 6×11
- 77 = 7×11
- 88 = 8×11

All are multiples of 11!

Also, notice:
- 55 → 5×11
- 66 → 6×11
- 77 → 7×11
- 88 → 8×11

So, they are all multiples of 11.

✔ Circled numbers are multiples of 11.

---

✔ Summary for Section A:

| Question | Answer |
|--------|--------|
| 1a) Shaded numbers (12,15,24,33,42,45) | Multiples of 3 |
| 1b) Circled numbers (12,24,32,44) | Multiples of 4 |
| 2a) Shaded numbers (55,60,...,90) | Multiples of 5 |
| 2b) Circled numbers (55,66,77,88) | Multiples of 11 |

---

Section B: Complete these sentences

Use the example:
> 3 is a factor of 9; 9 is a multiple of 3.

Let’s go one by one.

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#### 1) 18 is a ________ of 3; 3 is a ________ of 18.

- 18 ÷ 3 = 6 → so 18 is a multiple of 3
- 3 divides 18 → so 3 is a factor of 18

✔ Answer: 18 is a multiple of 3; 3 is a factor of 18.

---

#### 2) 4 and 5 are common _________ of 20; 20 is a ___________ of 4 and 5.

- 4 and 5 both divide 20 → so they are factors of 20
- 20 is divisible by both → so it's a common multiple of 4 and 5

✔ Answer: 4 and 5 are common factors of 20; 20 is a common multiple of 4 and 5.

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#### 3) The highest common _________ of 18 and 30 is 6.

- HCF (Highest Common Factor) of 18 and 30 is 6

✔ Answer: The highest common factor of 18 and 30 is 6.

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#### 4) 12 is a _________ of 24; 24 is a _________ of 12.

- 12 divides 24 → 12 is a factor of 24
- 24 = 12 × 2 → 24 is a multiple of 12

✔ Answer: 12 is a factor of 24; 24 is a multiple of 12.

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#### 5) The lowest common _________ of 4 and 10 is 20.

- LCM (Lowest Common Multiple) of 4 and 10 is 20

✔ Answer: The lowest common multiple of 4 and 10 is 20.

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#### 6) 1 is a _________ of every number.

- Every number divided by 1 is itself → so 1 divides every number → 1 is a factor of every number

✔ Answer: 1 is a factor of every number.

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#### 7) 0 is a _________ of every number.

- Any number multiplied by 0 is 0 → so 0 is a multiple of every number

Example: 5 × 0 = 0 → 0 is a multiple of 5

✔ Answer: 0 is a multiple of every number.

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✔ Final Answers:

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Section A:

1)
- Shaded numbers: They are all multiples of 3
- Circled numbers: They are all multiples of 4

2)
- Shaded numbers: They are all multiples of 5
- Circled numbers: They are all multiples of 11

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Section B:

1) 18 is a multiple of 3; 3 is a factor of 18.
2) 4 and 5 are common factors of 20; 20 is a common multiple of 4 and 5.
3) The highest common factor of 18 and 30 is 6.
4) 12 is a factor of 24; 24 is a multiple of 12.
5) The lowest common multiple of 4 and 10 is 20.
6) 1 is a factor of every number.
7) 0 is a multiple of every number.

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