Prime Factors: Stretch (Year 5) | CGP Plus - Free Printable

Problem Analysis:

The task involves working with prime factors and factor trees. Let's break it down step by step:

#### Part 1: Prime Numbers
We are given three prime numbers: 23, 31, and 47. The task is to complete the factor trees for these numbers and determine how many prime factors each has.

- Prime Numbers: A prime number has exactly two factors: 1 and itself. Since the question asks for prime factors, we only consider the number itself as a prime factor (since 1 is not a prime number).

##### Step-by-Step Solution for Prime Numbers:
1. 23:
- 23 is a prime number.
- Its only prime factor is 23.
- Factor tree:
```
23
/ \
1 23
```
- Number of prime factors: 1 (only 23 itself).

2. 31:
- 31 is a prime number.
- Its only prime factor is 31.
- Factor tree:
```
31
/ \
1 31
```
- Number of prime factors: 1 (only 31 itself).

3. 47:
- 47 is a prime number.
- Its only prime factor is 47.
- Factor tree:
```
47
/ \
1 47
```
- Number of prime factors: 1 (only 47 itself).

##### Explanation:
Each of the given numbers (23, 31, and 47) is a prime number. By definition, a prime number has exactly one prime factor, which is the number itself.

#### Part 2: Composite Numbers
We are given three composite numbers: 28, 42, and 45. The task is to complete the factor trees for these numbers and determine their prime factors.

##### Step-by-Step Solution for Composite Numbers:

1. 28:
- Start with 28. Break it down into factors:
- \( 28 = 4 \times 7 \)
- Further break down 4:
- \( 4 = 2 \times 2 \)
- The prime factors are: \( 2, 2, 7 \).
- Factor tree:
```
28
/ \
4 7
/ \
2 2
```
- Prime factorization: \( 28 = 2 \times 2 \times 7 \).

2. 42:
- Start with 42. Break it down into factors:
- \( 42 = 6 \times 7 \)
- Further break down 6:
- \( 6 = 2 \times 3 \)
- The prime factors are: \( 2, 3, 7 \).
- Factor tree:
```
42
/ \
6 7
/ \
2 3
```
- Prime factorization: \( 42 = 2 \times 3 \times 7 \).

3. 45:
- Start with 45. Break it down into factors:
- \( 45 = 9 \times 5 \)
- Further break down 9:
- \( 9 = 3 \times 3 \)
- The prime factors are: \( 3, 3, 5 \).
- Factor tree:
```
45
/ \
9 5
/ \
3 3
```
- Prime factorization: \( 45 = 3 \times 3 \times 5 \).

##### Explanation:
For composite numbers, we break them down into their prime factors using factor trees. Each branch represents a factorization step until all factors are prime numbers.

Final Answers:

1. Prime Numbers:
- 23: 1 prime factor (23).
- 31: 1 prime factor (31).
- 47: 1 prime factor (47).

2. Composite Numbers:
- 28: Prime factors are \( 2, 2, 7 \).
Factor tree:
```
28
/ \
4 7
/ \
2 2
```
- 42: Prime factors are \( 2, 3, 7 \).
Factor tree:
```
42
/ \
6 7
/ \
2 3
```
- 45: Prime factors are \( 3, 3, 5 \).
Factor tree:
```
45
/ \
9 5
/ \
3 3
```

Boxed Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Prime Numbers:} \\
23: 1 \text{ prime factor (23)} \\
31: 1 \text{ prime factor (31)} \\
47: 1 \text{ prime factor (47)} \\
\\
\text{Composite Numbers:} \\
28: 2 \times 2 \times 7 \\
42: 2 \times 3 \times 7 \\
45: 3 \times 3 \times 5 \\
\end{array}
}
\]