Multiples and Factors worksheet | Factors and multiples ... - Free Printable

Problem Solving and Explanation

Let's solve the worksheet step by step.

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#### 1. Complete the following table and write all the factors of 28.

The goal is to fill in the blanks in the multiplication equations that equal 28, and then list all the factors of 28.

##### Step 1: Fill in the table
We need to find pairs of numbers that multiply to give 28.

- \( 1 \times \_\_ = 28 \)
- \( 1 \times 28 = 28 \)

- \( \_\_ \times 14 = 28 \)
- \( 2 \times 14 = 28 \)

- \( 4 \times \_\_ = 28 \)
- \( 4 \times 7 = 28 \)

- \( 7 \times \_\_ = 28 \)
- \( 7 \times 4 = 28 \)

So, the completed table is:
\[
\begin{aligned}
&1 \times 28 = 28 \\
&2 \times 14 = 28 \\
&4 \times 7 = 28 \\
&7 \times 4 = 28
\end{aligned}
\]

##### Step 2: List all the factors of 28
From the table, we see the factor pairs are:
- \( (1, 28) \)
- \( (2, 14) \)
- \( (4, 7) \)

The factors of 28 are: \( 1, 2, 4, 7, 14, 28 \).

Answer for Question 1:
\[
\boxed{1, 2, 4, 7, 14, 28}
\]

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#### 2. (a) Match each number with its factors.

We need to match the given numbers with their correct sets of factors.

- Number 12: The factors of 12 are \( 1, 2, 3, 4, 6, 12 \).
- Number 16: The factors of 16 are \( 1, 2, 4, 8, 16 \).
- Number 20: The factors of 20 are \( 1, 2, 4, 5, 10, 20 \).

Now, let’s match them with the given options:
- \( 12 \) matches with \( 1, 2, 3, 4, 6, 12 \).
- \( 16 \) matches with \( 1, 2, 4, 8, 16 \).
- \( 20 \) matches with \( 1, 2, 4, 5, 10, 20 \).

Answer for Question 2(a):
\[
\boxed{
\begin{array}{c|c}
\text{Numbers} & \text{Factors} \\
\hline
12 & 1, 2, 3, 4, 6, 12 \\
16 & 1, 2, 4, 8, 16 \\
20 & 1, 2, 4, 5, 10, 20 \\
\end{array}
}
\]

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#### 2. (b) What is the HCF of 12, 16, and 20?

To find the Highest Common Factor (HCF) of 12, 16, and 20, we first list their factors:

- Factors of 12: \( 1, 2, 3, 4, 6, 12 \)
- Factors of 16: \( 1, 2, 4, 8, 16 \)
- Factors of 20: \( 1, 2, 4, 5, 10, 20 \)

The common factors of 12, 16, and 20 are: \( 1, 2, 4 \).

The highest common factor is \( 4 \).

Answer for Question 2(b):
\[
\boxed{4}
\]

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#### 3. Read the following statements and write (T) for True and (F) for False accordingly.

##### (a) No number can have ‘0’ as a factor.
- This is true because division by zero is undefined in mathematics. Therefore, no number can have 0 as a factor.
- Answer: \( \boxed{T} \)

##### (b) 11 and 13 are composite numbers.
- This is false because 11 and 13 are prime numbers. A composite number has more than two factors, but 11 and 13 only have two factors: 1 and themselves.
- Answer: \( \boxed{F} \)

##### (c) The only even prime number is 2.
- This is true because 2 is the smallest prime number and the only even prime number. All other even numbers are divisible by 2 and hence not prime.
- Answer: \( \boxed{T} \)

##### (d) 39 is the multiple of 8.
- This is false because 39 is not divisible by 8. When we divide 39 by 8, we get a remainder.
- Answer: \( \boxed{F} \)

##### (e) The smallest factor that a number can have is 1.
- This is true because every number is divisible by 1, and 1 is the smallest positive integer.
- Answer: \( \boxed{T} \)

Answer for Question 3:
\[
\boxed{T, F, T, F, T}
\]

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#### 4. Complete these factor trees and find the HCF of 45 and 75.

##### Step 1: Prime factorization of 45
We start with 45 and break it down into prime factors:
\[
45 \div 3 = 15 \quad \text{(3 is a prime factor)}
\]
\[
15 \div 3 = 5 \quad \text{(3 is a prime factor again)}
\]
\[
5 \div 5 = 1 \quad \text{(5 is a prime factor)}
\]

So, the prime factorization of 45 is:
\[
45 = 3 \times 3 \times 5 = 3^2 \times 5
\]

##### Step 2: Prime factorization of 75
We start with 75 and break it down into prime factors:
\[
75 \div 3 = 25 \quad \text{(3 is a prime factor)}
\]
\[
25 \div 5 = 5 \quad \text{(5 is a prime factor)}
\]
\[
5 \div 5 = 1 \quad \text{(5 is a prime factor again)}
\]

So, the prime factorization of 75 is:
\[
75 = 3 \times 5 \times 5 = 3 \times 5^2
\]

##### Step 3: Find the HCF of 45 and 75
The HCF is found by taking the lowest power of each common prime factor:
- Common prime factors of 45 and 75 are \( 3 \) and \( 5 \).
- The lowest power of 3 is \( 3^1 \).
- The lowest power of 5 is \( 5^1 \).

Thus, the HCF is:
\[
\text{HCF} = 3^1 \times 5^1 = 3 \times 5 = 15
\]

##### Step 4: Fill in the factor trees
- For 45:
\[
45 \rightarrow 3 \times 15 \rightarrow 3 \times (3 \times 5)
\]
So, the factor tree is:
\[
45 \rightarrow 3, 15 \rightarrow 3, 5
\]

- For 75:
\[
75 \rightarrow 3 \times 25 \rightarrow 3 \times (5 \times 5)
\]
So, the factor tree is:
\[
75 \rightarrow 3, 25 \rightarrow 3, 5, 5
\]

##### Step 5: Write the answers
- Prime factorization of 45: \( 3 \times 3 \times 5 \)
- Prime factorization of 75: \( 3 \times 5 \times 5 \)
- Product of common factors: \( 3 \times 5 = 15 \)
- HCF of 45 and 75: \( 15 \)

Answer for Question 4:
\[
\boxed{15}
\]

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Final Answers:

1. \(\boxed{1, 2, 4, 7, 14, 28}\)
2. (a) \(\boxed{
\begin{array}{c|c}
\text{Numbers} & \text{Factors} \\
\hline
12 & 1, 2, 3, 4, 6, 12 \\
16 & 1, 2, 4, 8, 16 \\
20 & 1, 2, 4, 5, 10, 20 \\
\end{array}
}\)
(b) \(\boxed{4}\)
3. \(\boxed{T, F, T, F, T}\)
4. \(\boxed{15}\)