Factor Tree Worksheet 3: Fill in the missing numbers in the prime factor trees with prime factors in the circles.

Let's solve Factor Tree Worksheet 3 step by step. The goal is to complete each factor tree so that the final prime factors are shown at the bottom (leaves of the tree). We'll break down each number into its prime factors using factor trees.

---

1) 77

We start with 77.
- 77 ÷ 7 = 11
- Both 7 and 11 are prime numbers.

So:
```
77
/ \
7 11
```

✔ Prime factors: 7 × 11

---

2) 85

Break down 85:
- 85 ÷ 5 = 17
- Both 5 and 17 are prime.

So:
```
85
/ \
5 17
```

✔ Prime factors: 5 × 17

---

3) 69

Break down 69:
- 69 ÷ 3 = 23
- 3 and 23 are both prime.

So:
```
69
/ \
3 23
```

✔ Prime factors: 3 × 23

---

4) 52

Break down 52:
- 52 ÷ 2 = 26
- 26 ÷ 2 = 13
- 13 is prime.

So:
```
52
/ \
2 26
/ \
2 13
```

✔ Prime factors: 2 × 2 × 13 → 2² × 13

---

5) 48

Break down 48:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is prime.

So:
```
48
/ \
2 24
/ \
2 12
/ \
2 6
/ \
2 3
```

But we can simplify this as a balanced tree:

```
48
/ \
2 24
/ \
2 12
/ \
2 6
/ \
2 3
```

Or better, split differently:
- 48 = 6 × 8
- 6 = 2 × 3
- 8 = 2 × 4 = 2 × 2 × 2

But since the worksheet shows two branches from 48, we’ll use:
```
48
/ \
6 8
/ \ / \
2 3 2 4
/ \
2 2
```

But to keep it simple and correct, let’s do:
```
48
/ \
2 24
/ \
2 12
/ \
2 6
/ \
2 3
```

So final prime factors: 2 × 2 × 2 × 2 × 3 → 2⁴ × 3

✔ Prime factors: 2, 2, 2, 2, 3

---

6) 105

Break down 105:
- 105 ÷ 5 = 21
- 21 ÷ 3 = 7
- All three are prime.

So:
```
105
/ \
5 21
/ \
3 7
```

✔ Prime factors: 5 × 3 × 7

---

7) 120

Break down 120:
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5

So:
```
120
/ \
2 60
/ \
2 30
/ \
2 15
/ \
3 5
```

✔ Prime factors: 2 × 2 × 2 × 3 × 5 → 2³ × 3 × 5

---

8) 88

Break down 88:
- 88 ÷ 2 = 44
- 44 ÷ 2 = 22
- 22 ÷ 2 = 11
- 11 is prime.

So:
```
88
/ \
2 44
/ \
2 22
/ \
2 11
```

✔ Prime factors: 2 × 2 × 2 × 11 → 2³ × 11

---

9) 90

Break down 90:
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5

So:
```
90
/ \
2 45
/ \
3 15
/ \
3 5
```

✔ Prime factors: 2 × 3 × 3 × 5 → 2 × 3² × 5

---

10) 102

Break down 102:
- 102 ÷ 2 = 51
- 51 ÷ 3 = 17
- 17 is prime.

So:
```
102
/ \
2 51
/ \
3 17
```

✔ Prime factors: 2 × 3 × 17

---

✔ Final Answers:

| Number | Prime Factorization |
|--------|---------------------|
| 77 | 7 × 11 |
| 85 | 5 × 17 |
| 69 | 3 × 23 |
| 52 | 2 × 2 × 13 |
| 48 | 2 × 2 × 2 × 2 × 3 |
| 105 | 3 × 5 × 7 |
| 120 | 2 × 2 × 2 × 3 × 5 |
| 88 | 2 × 2 × 2 × 11 |
| 90 | 2 × 3 × 3 × 5 |
| 102 | 2 × 3 × 17 |

---

Now, fill in the blanks on the worksheet accordingly. For example:

1) 77 → 7 and 11
2) 85 → 5 and 17
3) 69 → 3 and 23
4) 52 → 2 and 26 → 2 and 13
5) 48 → 6 and 8 → (6 → 2,3), (8 → 2,4 → 2,2,2) → but in tree format:
- 48 → 2 and 24 → 2 and 12 → 2 and 6 → 2 and 3
So leaves: 2, 2, 2, 2, 3
6) 105 → 5 and 21 → 3 and 7
7) 120 → 2 and 60 → 2 and 30 → 2 and 15 → 3 and 5
8) 88 → 2 and 44 → 2 and 22 → 2 and 11
9) 90 → 2 and 45 → 3 and 15 → 3 and 5
10) 102 → 2 and 51 → 3 and 17

Each tree ends in prime numbers — that’s the goal!

Let me know if you'd like a visual version or printable answers!