Factor Tree Worksheet 6 Answers showing prime factorization of four numbers with completed factor trees and solutions.

The image you've uploaded is a Factor Tree Worksheet 6 Answers page, which shows the prime factorization of four numbers using factor trees. Let's go through each problem step by step and explain how the factor trees work and why they lead to the correct prime factorizations.

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🔍 What is a Factor Tree?

A factor tree is a diagram used to break down a number into its prime factors. You start with the original number and split it into two factors, then continue breaking down each composite factor until only prime numbers remain at the bottom.

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## ✔ Problem 1: 68

```
68
/ \
2 34
/ \
2 17
```

- 68 ÷ 2 = 34 → 2 is prime ✔
- 34 ÷ 2 = 17 → 2 is prime ✔
- 17 is a prime number ✔

So:
> 68 = 2 × 2 × 17 = 2² × 17

✔ Correct!

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## ✔ Problem 2: 104

```
104
/ \
4 26
/ \ / \
2 2 2 13
```

- 104 ÷ 4 = 26 → 4 and 26 are both composite
- 4 = 2 × 2 → both primes ✔
- 26 = 2 × 13 → 2 and 13 are primes ✔

So:
> 104 = 2 × 2 × 2 × 13 = 2³ × 13

✔ Correct!

> Note: You could also split 104 as 2 × 52 or 8 × 13 — different paths, same result.

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## ✔ Problem 3: 78

```
78
/ \
2 39
/ \
3 13
```

- 78 ÷ 2 = 39 → 2 is prime ✔
- 39 ÷ 3 = 13 → 3 and 13 are both primes ✔

So:
> 78 = 2 × 3 × 13

✔ Correct!

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## ✔ Problem 4: 225

```
225
/ \
9 25
/ \ / \
3 3 5 5
```

- 225 ÷ 9 = 25 → both 9 and 25 are composite
- 9 = 3 × 3 → both primes ✔
- 25 = 5 × 5 → both primes ✔

So:
> 225 = 3 × 3 × 5 × 5 = 3² × 5²

✔ Correct!

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## 🧠 Key Notes from the Worksheet:

- Prime factorization is unique (Fundamental Theorem of Arithmetic), so regardless of the path taken in the factor tree, the final prime factors will be the same.
- The order of the factors doesn't matter — just the set of prime factors.
- Composite numbers are broken down until only primes remain.

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✔ Final Answers Summary:

| Number | Prime Factorization |
|--------|---------------------|
| 68 | 2² × 17 |
| 104 | 2³ × 13 |
| 78 | 2 × 3 × 13 |
| 225 | 3² × 5² |

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Let me know if you'd like help creating your own factor trees or want practice problems!