Factor Tree Worksheet for Prime Factorization Practice

Let's solve the Factor Tree Worksheet step by step. The goal is to break down each number into its prime factors using a factor tree, and then fill in the apples at the bottom with the prime factors.

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🔹 Understanding the Example:

The example shows:
- 100
- 100 → 2 × 50
- 50 → 2 × 25
- 25 → 5 × 5
- So prime factors are: 2, 2, 5, 5
- Apples show: 2, 2, 5, 5

So we do the same for the other numbers.

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## ✔ Problem 1: 54

We start factoring 54:

1. 54 = 2 × 27 (since 54 is even)
2. 27 = 3 × 9
3. 9 = 3 × 3

So the full factor tree:
```
54
/ \
2 27
/ \
3 9
/ \
3 3
```

Prime factors: 2, 3, 3, 3

✔ Apples should be: 2, 3, 3, 3

---

## ✔ Problem 2: 81

81 is odd, not divisible by 2.

1. 81 = 9 × 9
2. 9 = 3 × 3, so:
- 81 = 3 × 3 × 3 × 3

Or:
```
81
/ \
9 9
/ \ / \
3 3 3 3
```

Prime factors: 3, 3, 3, 3

✔ Apples: 3, 3, 3, 3

---

## ✔ Problem 3: 72

Break down 72:

1. 72 = 8 × 9
- Or better: 72 = 2 × 36
2. 36 = 6 × 6 → or 36 = 2 × 18
- Let’s go step by step:
- 72 → 2 × 36
- 36 → 2 × 18
- 18 → 2 × 9
- 9 → 3 × 3

So:
```
72
/ \
2 36
/ \
2 18
/ \
2 9
/ \
3 3
```

Prime factors: 2, 2, 2, 3, 3

✔ Apples: 2, 2, 2, 3, 3

---

## ✔ Problem 4: 450

Start factoring 450:

1. 450 = 2 × 225 (even number)
2. 225 = 15 × 15
- 15 = 3 × 5
- So 225 = 3 × 5 × 3 × 5 = 3² × 5²

So:
```
450
/ \
2 225
/ \
15 15
/ \ / \
3 5 3 5
```

But let’s make it cleaner:
- 450 → 2 × 225
- 225 → 15 × 15 → but better to split as:
- 225 → 3 × 75
- 75 → 3 × 25
- 25 → 5 × 5

Better path:
```
450
/ \
2 225
/ \
3 75
/ \
3 25
/ \
5 5
```

So prime factors: 2, 3, 3, 5, 5

✔ Apples: 2, 3, 3, 5, 5

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🍎 Final Answers (Fill in the Apples):

#### 🔹 54 → Apples: 2, 3, 3, 3

#### 🔹 81 → Apples: 3, 3, 3, 3

#### 🔹 72 → Apples: 2, 2, 2, 3, 3

#### 🔹 450 → Apples: 2, 3, 3, 5, 5

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✔ Summary of Prime Factorizations:

| Number | Prime Factors |
|--------|----------------|
| 54 | 2 × 3³ |
| 81 | 3⁴ |
| 72 | 2³ × 3² |
| 450 | 2 × 3² × 5² |

You can now fill the blanks in the trees accordingly. Each apple holds one prime factor, and you list them all at the bottom.

Let me know if you'd like a visual version of the completed trees!