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 = 6

So, the other factor is 6.

Now, break down 6 into two factors:
- 6 = 2 × 3 (both prime)

So, the complete tree is:

```
30
/ \
5 6
/ \
2 3
```

✔ Prime factors of 30: 2 × 3 × 5

---

🔹 2. Factor Tree for 81

We are given:
```
81
/ \
? 9
```

We know:
- 81 ÷ 9 = 9

So, the missing factor is 9.

Now, break down both 9s:
- 9 = 3 × 3

So, each 9 splits into:
```
9
/ \
3 3
```

Full tree:
```
81
/ \
9 9
/ \ / \
3 3 3 3
```

✔ Prime factors of 81: 3 × 3 × 3 × 3 = 3⁴

---

🔹 3. Factor Tree for 56

We are given:
```
56
/ \
? ?
```

Let’s pick a factor pair. Since 56 is even, divide by 2:
- 56 ÷ 2 = 28 → so first branch: 2 and 28

Now, split 28:
- 28 = 2 × 14

Then split 14:
- 14 = 2 × 7

So the full tree:
```
56
/ \
2 28
/ \
2 14
/ \
2 7
```

But in the worksheet, there are only three levels, and the second level has one blank box and then a smaller tree below it.

Let’s match the structure:

Given:
```
56
/ \
○ □
/ \
○ □
/ \
○ ○
```

So:
- First split: 56 = ? × ?
- We can use 7 and 8 (since 56 = 7 × 8), but 7 is prime, and 8 is not.

Better: Use 8 and 7? But let's go with standard approach.

Wait — maybe they expect 56 = 7 × 8?

But 8 is not prime.

Alternatively, 56 = 2 × 28 → then 28 = 4 × 7 → 4 = 2×2

But we need to fill the boxes correctly.

Looking at the diagram:
- Top: 56
- Two branches: left circle (unknown), right box (unknown)
- Then the right box branches into two circles
- One of those circles branches further into two more circles

So likely:
- 56 = 7 × 8 → 7 is prime, so left circle = 7
- Right box = 8
- Then 8 = 2 × 4
- Then 4 = 2 × 2

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

But wait — 8 is not prime, so we need to keep breaking it down.

Yes, that works.

So the filled tree:
```
56
/ \
7 8
/ \
2 4
/ \
2 2
```

But the diagram shows:
- Left circle: should be 7
- Right box: 8
- Then from 8: two circles → 2 and 4
- From 4: two circles → 2 and 2

So yes, this fits.

✔ Prime factors: 7 × 2 × 2 × 2 = 2³ × 7

---

🔹 4. Factor Tree for 72

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

We need to split 72 into two factors. Let’s use 8 and 9:
- 72 = 8 × 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
```

But the diagram has:
- Two boxes at the top level
- Each splits into two circles
- One of the bottom circles splits into two more circles

So likely:
- 72 = 8 × 9 → left box = 8, right box = 9
- 8 → 2 and 4 → 4 → 2 and 2
- 9 → 3 and 3

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

This matches the structure.

✔ Prime factors: 2 × 2 × 2 × 3 × 3 = 2³ × 3²

---

✔ Final Answers (Fill in the blanks):

#### 1. 30
```
30
/ \
5 6
/ \
2 3
```

- Missing numbers: 6, 2, 3

#### 2. 81
```
81
/ \
9 9
/ \ / \
3 3 3 3
```

- Missing numbers: 9, 3, 3, 3, 3

#### 3. 56
```
56
/ \
7 8
/ \
2 4
/ \
2 2
```

- Missing numbers: 7, 8, 2, 4, 2, 2

#### 4. 72
```
72
/ \
8 9
/ \ / \
2 4 3 3
/ \
2 2
```

- Missing numbers: 8, 9, 2, 4, 2, 2, 3, 3

---

📌 Summary of Prime Factors:

| Number | Prime Factorization |
|--------|---------------------|
| 30 | 2 × 3 × 5 |
| 81 | 3⁴ |
| 56 | 2³ × 7 |
| 72 | 2³ × 3² |

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