Grade 4 math worksheet on multiples and factors, including exercises on finding multiples, checking multiples, common multiples, LCM, factors using multiplication and division, and HCF.

Problem: Solving the "Multiples and Factors" Worksheet

Below is a detailed solution to each part of the worksheet. Let's go through it step by step.

---

#### 1. Write the first five multiples of the following.
- a. 4
- Multiples of 4 are obtained by multiplying 4 by the natural numbers (1, 2, 3, 4, 5).
- Calculation:
\[
4 \times 1 = 4, \quad 4 \times 2 = 8, \quad 4 \times 3 = 12, \quad 4 \times 4 = 16, \quad 4 \times 5 = 20
\]
- First five multiples of 4: 4, 8, 12, 16, 20

- b. 11
- Multiples of 11 are obtained by multiplying 11 by the natural numbers (1, 2, 3, 4, 5).
- Calculation:
\[
11 \times 1 = 11, \quad 11 \times 2 = 22, \quad 11 \times 3 = 33, \quad 11 \times 4 = 44, \quad 11 \times 5 = 55
\]
- First five multiples of 11: 11, 22, 33, 44, 55

---

#### 2. Answer the following in Yes or No.
- a. Is 112 a multiple of 4?
- To check if 112 is a multiple of 4, divide 112 by 4:
\[
112 \div 4 = 28
\]
- Since the result is a whole number, 112 is a multiple of 4.
- Answer: Yes

- b. Is 1586 a multiple of 15?
- To check if 1586 is a multiple of 15, divide 1586 by 15:
\[
1586 \div 15 = 105.7333\ldots
\]
- Since the result is not a whole number, 1586 is not a multiple of 15.
- Answer: No

---

#### 3. Find four common multiples of the following.
- a. 4 and 5
- First, find the least common multiple (LCM) of 4 and 5.
- Prime factorization:
\[
4 = 2^2, \quad 5 = 5^1
\]
- LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(4, 5) = 2^2 \times 5 = 20
\]
- Common multiples of 4 and 5 are multiples of 20:
\[
20, 40, 60, 80, \ldots
\]
- Four common multiples: 20, 40, 60, 80

- b. 3 and 6
- First, find the LCM of 3 and 6.
- Prime factorization:
\[
3 = 3^1, \quad 6 = 2^1 \times 3^1
\]
- LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(3, 6) = 2^1 \times 3^1 = 6
\]
- Common multiples of 3 and 6 are multiples of 6:
\[
6, 12, 18, 24, \ldots
\]
- Four common multiples: 6, 12, 18, 24

- c. 5 and 10
- First, find the LCM of 5 and 10.
- Prime factorization:
\[
5 = 5^1, \quad 10 = 2^1 \times 5^1
\]
- LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(5, 10) = 2^1 \times 5^1 = 10
\]
- Common multiples of 5 and 10 are multiples of 10:
\[
10, 20, 30, 40, \ldots
\]
- Four common multiples: 10, 20, 30, 40

---

#### 4. Find the LCM of the following.
- a. 6 and 8
- Prime factorization:
\[
6 = 2^1 \times 3^1, \quad 8 = 2^3
\]
- LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(6, 8) = 2^3 \times 3^1 = 8 \times 3 = 24
\]
- LCM: 24

- b. 5 and 7
- Prime factorization:
\[
5 = 5^1, \quad 7 = 7^1
\]
- LCM is the product of the highest powers of all prime factors:
\[
\text{LCM}(5, 7) = 5^1 \times 7^1 = 5 \times 7 = 35
\]
- LCM: 35

---

#### 5. Find the factors of the following number using multiplication method.
- a. 25
- Factors are numbers that multiply to give 25. We list pairs of factors:
\[
1 \times 25 = 25, \quad 5 \times 5 = 25
\]
- Factors of 25: 1, 5, 25

---

#### 6. Find the factors of the following number using division method.
- a. 72
- Divide 72 by all numbers from 1 to 72 to find the factors:
\[
72 \div 1 = 72, \quad 72 \div 2 = 36, \quad 72 \div 3 = 24, \quad 72 \div 4 = 18
\]
\[
72 \div 6 = 12, \quad 72 \div 8 = 9, \quad 72 \div 9 = 8, \quad 72 \div 12 = 6
\]
\[
72 \div 18 = 4, \quad 72 \div 24 = 3, \quad 72 \div 36 = 2, \quad 72 \div 72 = 1
\]
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

---

#### 7. Find the HCF of the following.
- a. 8 and 12
- Prime factorization:
\[
8 = 2^3, \quad 12 = 2^2 \times 3^1
\]
- HCF is the product of the lowest powers of common prime factors:
\[
\text{HCF}(8, 12) = 2^2 = 4
\]
- HCF: 4

---

Final Answers:

1.
- a. 4, 8, 12, 16, 20
- b. 11, 22, 33, 44, 55

2.
- a. Yes
- b. No

3.
- a. 20, 40, 60, 80
- b. 6, 12, 18, 24
- c. 10, 20, 30, 40

4.
- a. 24
- b. 35

5.
- a. 1, 5, 25

6.
- a. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

7.
- a. 4

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
&1. \text{a. } 4, 8, 12, 16, 20; \text{ b. } 11, 22, 33, 44, 55 \\
&2. \text{a. Yes; b. No} \\
&3. \text{a. } 20, 40, 60, 80; \text{ b. } 6, 12, 18, 24; \text{ c. } 10, 20, 30, 40 \\
&4. \text{a. } 24; \text{ b. } 35 \\
&5. \text{a. } 1, 5, 25 \\
&6. \text{a. } 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 \\
&7. \text{a. } 4
\end{aligned}
}
\]