Factors and Multiples interactive exercise - 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 result in 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.

- \( 1 \times \_\_ = 28 \)
- The missing number is \( 28 \div 1 = 28 \).
- So, \( 1 \times 28 = 28 \).

- \( \_\_ \times 14 = 28 \)
- The missing number is \( 28 \div 14 = 2 \).
- So, \( 2 \times 14 = 28 \).

- \( 4 \times \_\_ = 28 \)
- The missing number is \( 28 \div 4 = 7 \).
- So, \( 4 \times 7 = 28 \).

- \( 7 \times \_\_ = 28 \)
- The missing number is \( 28 \div 7 = 4 \).
- So, \( 7 \times 4 = 28 \).

##### Completed Table:
\[
\begin{array}{|c|}
\hline
1 \times 28 = 28 \\
2 \times 14 = 28 \\
4 \times 7 = 28 \\
7 \times 4 = 28 \\
\hline
\end{array}
\]

##### Step 2: List all the factors of 28
From the table, the factors of 28 are:
\[ 1, 2, 4, 7, 14, 28 \]

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

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

We need to match each number (12, 16, 20) with its correct set of factors.

##### Step 1: Factors of 12
The factors of 12 are the numbers that divide 12 evenly:
\[ 1, 2, 3, 4, 6, 12 \]

##### Step 2: Factors of 16
The factors of 16 are the numbers that divide 16 evenly:
\[ 1, 2, 4, 8, 16 \]

##### Step 3: Factors of 20
The factors of 20 are the numbers that divide 20 evenly:
\[ 1, 2, 4, 5, 10, 20 \]

##### Matching:
- \( 12 \) matches with \( 1, 2, 3, 4, 6, 12 \)
- \( 16 \) matches with \( 1, 2, 4, 8, 16 \)
- \( 20 \) matches with \( 1, 2, 4, 5, 10, 20 \)

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

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

The Highest Common Factor (HCF) is the largest number that divides all three numbers (12, 16, and 20) without leaving a remainder.

##### Step 1: Find the factors of each number
- Factors of 12: \( 1, 2, 3, 4, 6, 12 \)
- Factors of 16: \( 1, 2, 4, 8, 16 \)
- Factors of 20: \( 1, 2, 4, 5, 10, 20 \)

##### Step 2: Identify the common factors
The common factors of 12, 16, and 20 are:
\[ 1, 2, 4 \]

##### Step 3: Determine the highest common factor
The highest common factor is:
\[ 4 \]

##### Final Answer for Part 2(b):
\[
\boxed{4}
\]

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

##### Statement (a): No number can have ‘0’ as a factor.
- True: By definition, a factor of a number must divide it evenly, and division by 0 is undefined. Therefore, 0 cannot be a factor of any number.
- Answer: \( T \)

##### Statement (b): 11 and 13 are composite numbers.
- False: 11 and 13 are both prime numbers because they have exactly two distinct positive divisors: 1 and themselves.
- Answer: \( F \)

##### Statement (c): The only even prime number is 2.
- True: By definition, a prime number has exactly two distinct positive divisors: 1 and itself. The only even number that satisfies this condition is 2.
- Answer: \( T \)

##### Statement (d): 39 is the multiple of 8.
- False: To check if 39 is a multiple of 8, we divide 39 by 8:
\[
39 \div 8 = 4.875
\]
Since the result is not an integer, 39 is not a multiple of 8.
- Answer: \( F \)

##### Statement (e): The smallest factor that a number can have is 1.
- True: By definition, every number is divisible by 1, and 1 is the smallest positive integer.
- Answer: \( T \)

##### Final Answers for Part 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: Write the prime factorizations
\[
45 = 3^2 \times 5
\]
\[
75 = 3 \times 5^2
\]

##### Step 4: Identify the common factors
The common prime factors of 45 and 75 are:
- \( 3 \) (appears once in both)
- \( 5 \) (appears once in both)

##### Step 5: Calculate the HCF
The HCF is the product of the lowest powers of the common prime factors:
\[
\text{HCF} = 3^1 \times 5^1 = 3 \times 5 = 15
\]

##### Final Answers for Part 4:
\[
\boxed{15}
\]

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Final Summary of All Answers:

1. Factors of 28: \(\boxed{1, 2, 4, 7, 14, 28}\)
2. (a) Matching:
\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Numbers} & \text{Factors} \\
\hline
12 & 1, 2, 3, 4, 6, 12 \\
16 & 1, 2, 4, 8, 16 \\
20 & 1, 2, 4, 5, 10, 20 \\
\hline
\end{array}
}
\]
(b) HCF of 12, 16, and 20: \(\boxed{4}\)
3. Statements: \(\boxed{T, F, T, F, T}\)
4. HCF of 45 and 75: \(\boxed{15}\)