Prime factor trees worksheets| Worsheets library - Free Printable

Let's solve each of the prime factor trees step by step. The goal is to break down each number into its prime factors using a factor tree.

---

🔹 1. Factor Tree for 30

We are given:
```
30
/ \
5 ?
```

We know that:
- $ 30 = 5 \times ? $
- $ 30 ÷ 5 = 6 $

So, the missing number is 6.

Now, break down 6:
- $ 6 = 2 \times 3 $ → both are prime

So, the completed tree:
```
30
/ \
5 6
/ \
2 3
```

✔ Prime factors of 30: 2 × 3 × 5

---

🔹 2. Factor Tree for 81

Given:
```
81
/ \
? 9
```

We know:
- $ 81 ÷ 9 = 9 $
- So, $ 81 = 9 × 9 $

But wait — this isn’t correct because we want prime factors.

Wait — actually, let’s double-check:
- $ 81 = 9 × 9 $? No! $ 9 × 9 = 81 $, yes, but we need to start with a prime factor.

But here, we’re given one branch as 9, so we must have:
- $ 81 = ? × 9 $
- $ 81 ÷ 9 = 9 $

So the missing number is 9.

So the top level:
```
81
/ \
9 9
```

Now break down each 9:
- $ 9 = 3 × 3 $

So each 9 becomes:
```
9
/ \
3 3
```

So full tree:
```
81
/ \
9 9
/ \ / \
3 3 3 3
```

✔ Prime factors of 81: 3 × 3 × 3 × 3 = $ 3^4 $

---

🔹 3. Factor Tree for 56

Given:
```
56
/ \
? ?
```

We can choose any factor pair. Let’s pick:
- $ 56 = 7 × 8 $ (since 7 is prime)

So:
```
56
/ \
7 8
```

Now break down 8:
- $ 8 = 2 × 4 $
- $ 4 = 2 × 2 $

So:
```
56
/ \
7 8
/ \
2 4
/ \
2 2
```

But in the diagram, there’s only one box below 56, then one box below that, and two boxes under it. So structure:
```
56
/ \
? ?
/ \
? ?
/ \
? ?
```

Let’s fill in:

- First split: $ 56 = 7 × 8 $ → so left = 7, right = 8
- Then split 8: $ 8 = 2 × 4 $
- Then split 4: $ 4 = 2 × 2 $

So:
```
56
/ \
7 8
/ \
2 4
/ \
2 2
```

But the diagram has:
- One blank under 56 (left)
- One blank under 56 (right)
- Then one blank under the right branch
- Then two blanks under that

Wait — actually, looking at the layout:

```
56
/ \
○ □
|
□
/ \
○ ○
```

So likely:
- 56 splits into 7 and 8
- 8 splits into 2 and 4
- 4 splits into 2 and 2

So:
- Left circle: 7
- Right square: 8
- Below 8: 4
- Below 4: 2 and 2

✔ Prime factors of 56: 7 × 2 × 2 × 2 = $ 2^3 × 7 $

---

🔹 4. Factor Tree for 72

Given:
```
72
/ \
? ?
```

Let’s pick a factor pair. Good choice: $ 72 = 8 × 9 $

So:
- Left: 8
- Right: 9

Now:
- $ 8 = 2 × 4 $, $ 4 = 2 × 2 $ → so $ 8 = 2 × 2 × 2 $
- $ 9 = 3 × 3 $

So:
```
72
/ \
8 9
/ \ / \
2 4 3 3
/ \
2 2
```

Now check the diagram:
- Two branches from 72 → both filled
- Left branch (8) splits into 2 and 4
- 4 splits into 2 and 2
- Right branch (9) splits into 3 and 3

So:
- Top left: 8
- Top right: 9
- Under 8: 2 and 4
- Under 4: 2 and 2
- Under 9: 3 and 3

✔ Prime factors of 72: 2 × 2 × 2 × 3 × 3 = $ 2^3 × 3^2 $

---

✔ Final Answers:

#### 1. 30
```
30
/ \
5 6
/ \
2 3
```
Prime factors: 2 × 3 × 5

#### 2. 81
```
81
/ \
9 9
/ \ / \
3 3 3 3
```
Prime factors: 3⁴

#### 3. 56
```
56
/ \
7 8
/ \
2 4
/ \
2 2
```
Prime factors: 2³ × 7

#### 4. 72
```
72
/ \
8 9
/ \ / \
2 4 3 3
/ \
2 2
```
Prime factors: 2³ × 3²

---

📝 Summary of Fill-ins:

| Number | Missing Values |
|--------|----------------|
| 30 | 6, then 2 and 3 |
| 81 | 9, then 3,3 and 3,3 |
| 56 | 7 and 8; then 4; then 2 and 2 |
| 72 | 8 and 9; then 2,4 and 3,3; then 2,2 |

You can now fill in the worksheet accordingly.

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