Prime Factors Worksheet - Fill in the gaps and find the product of prime factors using factor trees.

Let's solve the Prime Factors worksheet step by step.

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Part 1: Fill in the gaps in each prime factor tree

We'll go through each lettered tree and fill in the missing numbers. The goal is to break down each number into its prime factors using a factor tree.

---

#### a. 20
```
20
/ \
4 5
/ \
2 2
```
- 4 = 2 × 2 → already given
- So, left branch: 4 splits into 2 and 2
- Right: 5 is already prime

✔ Completed:
```
20
/ \
4 5
/ \
2 2
```

---

#### b. 12
```
12
/ \
2 6
/ \
2 ?
```
- 6 = 2 × 3 → so missing number is 3

✔ Completed:
```
12
/ \
2 6
/ \
2 3
```

---

#### c. 24
```
24
/ \
4 ?
/ \
2 2
```
- 24 ÷ 4 = 6 → so right branch is 6
- Now split 6: 6 = 2 × 3

✔ Completed:
```
24
/ \
4 6
/ \ / \
2 2 2 3
```

---

#### d. 40
```
40
/ \
8 ?
/ \
4 ?
/ \
2 2
```
- 40 ÷ 8 = 5 → so right branch is 5
- 8 = 4 × 2 → but 4 is already shown as 2×2
- So, 8 splits into 4 and 2 → 4 splits into 2 and 2

So:
- Left side: 8 → 4 → (2,2), and 4 → (2,2) → already filled
- But we need to complete the 8 → 4 and 2 → so missing number under 8 is 2

Wait, let’s re-draw:

```
40
/ \
8 5
/ \
4 2
/ \
2 2
```

✔ So missing numbers:
- Under 8: 2
- Under 40: 5

✔ Completed:
```
40
/ \
8 5
/ \
4 2
/ \
2 2
```

---

#### e. 56
```
56
/ \
? 28
/ \
4 ?
/ \
2 2
```
- 56 ÷ 28 = 2 → so left branch is 2
- 28 = 4 × 7 → so right branch of 28 is 7
- 4 = 2×2 → already given

✔ Completed:
```
56
/ \
2 28
/ \
4 7
/ \
2 2
```

---

#### f. 72
```
72
/ \
8 ?
/ \
? 2
/ \
? 2
```
- 72 ÷ 8 = 9 → so right branch is 9
- 8 = 4 × 2 → but 4 is not yet shown
- Wait, 8 → split into 4 and 2 → 4 → 2 and 2 → so:
- 8 → 4 and 2
- 4 → 2 and 2
- So first blank under 8: 4
- Then under 4: 2
- Right branch: 9 → split into 3 and 3 → but it shows one branch with 3 → so other must be 3

But current diagram:
```
72
/ \
8 ?
/ \
? 2
/ \
? 2
```
And on the right:
```
?
/ \
? 3
```

So:
- 72 ÷ 8 = 9 → right branch is 9
- 9 = 3 × 3 → so both branches are 3

Now for 8:
- 8 = 4 × 2 → so top left under 8 is 4
- 4 = 2 × 2 → so under 4: 2 and 2

But currently:
- 8 → ? and 2 → so ? is 4
- Then 4 → ? and 2 → so ? is 2

So:
- First blank under 8: 4
- Then under 4: 2
- Right branch: 9, then splits into 3 and 3

✔ Completed:
```
72
/ \
8 9
/ \ / \
4 2 3 3
/ \
2 2
```

---

#### g. ?
```
?
/ \
4 4
/ \ / \
2 2 2 2
```
- 4 × 4 = 16 → so top number is 16

✔ Answer: Top number is 16

---

#### h. ?
```
?
/ \
5 ?
/ \
2 3
```
- Right branch: 2 × 3 = 6
- So top number: 5 × 6 = 30

✔ Answer: Top number is 30

---

#### i. 75
```
75
/ \
? ?
/ \ / \
5 5 ? ?
```
- 75 = 3 × 25 or 5 × 15
- Since left side is 5 and 5 → that's 25
- So 75 = 25 × 3 → so branches: 25 and 3
- 25 = 5 × 5 → already shown

So:
- Left branch: 25
- Right branch: 3

✔ Completed:
```
75
/ \
25 3
/ \
5 5
```

---

#### j. 80
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```
- We know 80 = 16 × 5 → 16 is 2⁴
- But look at structure: left side ends with 5 → so left branch must be multiple of 5
- Left branch has 5 and something → so left branch is 5 × ? = ?

Let’s work bottom up.

Left side:
- Bottom: 2 and 2 → 4
- Above: 4 and 5 → 20
- So left branch is 20

Then right branch: 80 ÷ 20 = 40

Now 40 → split into two numbers → one is shown as ending with 2 and 2 → so likely 4 and 2?

But 40 = 8 × 5? Or 40 = 4 × 10?

Wait, right branch is split into two parts, one of which leads to 2 and 2 → so maybe 4 and 10?

But let's see:

Right branch:
- One path: ? → ? → 2 and 2 → so that’s 4
- Other path: ? → ? → but only one blank shown

Actually, the diagram:
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```

From bottom:
- Left side: 2×2=4 → then 4×5=20 → so left top is 20
- Then right top: 80 ÷ 20 = 40
- Now 40 → split into two factors → one path ends in 2 and 2 → so that’s 4 → so 40 = 4 × 10?
- But 10 = 2 × 5 → but no 5 shown → wait

Alternatively, 40 = 8 × 5 → but 8 = 2×2×2 → so maybe 8 and 5?

But diagram shows:
- Right branch: one path goes to 2 and 2 → so that’s 4 → so 40 = 4 × 10 → then 10 = 2 × 5 → so we need to show 2 and 5

But only one blank shown under right branch → probably:

- 40 → 4 and 10
- 4 → 2 and 2 (already shown)
- 10 → 2 and 5 → but no 5 shown → so missing numbers:
- Right top: 40
- Then under 40: 4 and 10
- Under 10: 2 and 5

But the diagram only has one blank under right branch → so perhaps:

Let’s assume:
- Right branch: 40 → 4 and 10
- 4 → 2 and 2 (given)
- 10 → 2 and 5 → so missing numbers:
- Under 40: 4 and 10
- Under 10: 2 and 5

But only one blank under 40 → so maybe:

Wait — the diagram shows:
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```

So:
- Left: ? → ? and 5 → and below that: 2 and 2 → so bottom-left: 2×2=4 → then 4×5=20 → so left top is 20
- Right: ? → ? and ? → and below: 2 and 2 → so bottom-right: 2×2=4 → so 40 = ? × 4 → so ? = 10
- Then 10 → 2 and 5 → so missing: 2 and 5

So:
- Right top: 40
- Then under 40: 4 and 10
- Under 10: 2 and 5

But the diagram only has one blank under right branch → so likely:

- Right branch: 40 → 4 and 10 → so blank is 10
- Then under 10: 2 and 5 → so blanks: 2 and 5

But only one blank shown under 40 → so maybe the diagram assumes you fill in only one number per gap.

Looking at the layout:
- Right branch: 40 → ? and ? → but only one blank shown → maybe typo?

Wait — actually, in the image, the right side has:
```
?
/ \
? ?
/ \
2 2
```
No — wait, the original says:
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```
So:
- Left: ? → ? and 5 → and ? → 2 and 2 → so left top is 20
- Right: ? → ? and ? → and ? → 2 and 2 → so right top is 40
- Then under 40: one blank → so 40 → ? and ? → but only one blank → probably means: 40 → 4 and 10 → so one blank is 4, other is 10

But there’s only one blank under 40 → so maybe it’s designed to have only one missing number?

Wait — no, in the diagram, both sides have two blanks under the second level.

Actually, looking carefully:

The right branch:
- Starts with ? (top of right)
- Then splits into ? and ? → but only one blank shown under the right branch → no, actually, it shows:
```
?
/ \
? ?
/ \
2 2
```
But the bottom is 2 and 2 → so that’s 4 → so the middle node is 4 → so the top of right is 40 → then 40 → 4 and 10 → so missing: 4 and 10

But only one blank is shown under the right branch → this might be a formatting issue.

Wait — actually, the diagram shows:
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```
So:
- Left: ? → ? and 5 → and ? → 2 and 2 → so ? (left top) = 20
- Right: ? → ? and ? → and ? → 2 and 2 → so ? (right top) = 40
- Then under 40: two blanks → so 40 = ? × ? → possibilities: 4 and 10, or 5 and 8, etc.
- But bottom right: 2 and 2 → so 4 → so one of the factors is 4 → so 40 = 4 × 10 → so other is 10
- Then 10 → 2 and 5 → so missing numbers:
- Right top: 40
- Under 40: 4 and 10
- Under 10: 2 and 5

But the diagram only shows one blank under the right branch → so maybe it's meant to be:

- Right branch: 40 → 4 and 10 → so fill in 10 (since 4 is already implied)

But the blank is under the 40 → so likely: 10

And under 10: 2 and 5

But only one blank shown → maybe the diagram has a typo.

Alternatively, since the left side has 5, and right side has 2 and 2 → maybe 80 = 20 × 4 → but 40 is better.

Wait — 80 = 20 × 4 → 20 = 4×5, 4 = 2×2 → same thing.

But 80 = 20 × 4 → so right branch is 4 → but 4 is not prime → so must split further.

So:
- 80 → 20 and 4
- 20 → 4 and 5 → 4 → 2×2
- 4 → 2×2

So:
- Left: 20 → 4 and 5 → 4 → 2×2
- Right: 4 → 2×2

So:
- Top: 80
- Left: 20
- Right: 4
- Then 20 → 4 and 5
- 4 → 2 and 2
- 4 → 2 and 2

But in diagram, left has 5 and 2,2 → so yes.

So:
- Left top: 20
- Right top: 4
- Under 20: 4 and 5
- Under 4: 2 and 2

But diagram shows:
- Left: ? → ? and 5 → and ? → 2 and 2 → so ? = 4
- So left: 20 → 4 and 5 → 4 → 2 and 2

So missing:
- Left top: 20
- Left middle: 4
- Right top: 4
- Right middle: ? → but only one blank shown → maybe ? is 4, and the other is implied?

This is confusing.

Wait — let’s re-express:

Given:
```
80
/ \
? ?
/ \ / \
? 5 ? ?
/ \
2 2
```

Bottom left: 2×2 = 4 → so above that: 4 and 5 → 4×5 = 20 → so left top is 20
- So left branch: 20 → 4 and 5 → 4 → 2×2

Then right branch: 80 ÷ 20 = 40 → so right top is 40
- Then 40 → ? and ? → and one path ends in 2 and 2 → so that’s 4 → so 40 = 4 × 10 → so other is 10
- Then 10 → 2 and 5 → so missing:
- Right top: 40
- Under 40: 4 and 10
- Under 10: 2 and 5

But diagram shows only one blank under right branch → so likely:
- Right top: 40
- Under 40: 4 and 10 → so fill in 10 (if 4 is already shown)
- But no 4 shown → so both blanks needed

Perhaps the diagram intends:
- Right branch: 40 → 4 and 10 → so fill in 4 and 10
- Then under 10: 2 and 5

But only one blank shown under 40 → so maybe it's a mistake.

Alternatively, perhaps the right branch is 40 → 8 and 5 → 8 = 2×2×2 → but 8 = 4×2 → 4 = 2×2

But diagram shows only one 2,2 pair → so likely not.

Best guess:
- Left top: 20
- Right top: 40
- Under 40: 4 and 10
- Under 10: 2 and 5

So fill in:
- Left top: 20
- Left middle: 4
- Right top: 40
- Right middle: 10
- Then under 10: 2 and 5

But only one blank under 40 → so maybe the answer is 10

We’ll assume the missing number under 40 is 10

But let’s move on.

---

#### k. 36
```
36
/ \
? ?
/ \
3 ?
/ \
2 2
```
- Left branch: 3 × ? → and ? → 2×2 = 4 → so 3 × 4 = 12 → so left top is 12
- Then 36 ÷ 12 = 3 → so right top is 3
- But right branch has no numbers → so 3 is prime → so right top is 3
- But 36 = 12 × 3 → yes
- Then 12 = 3 × 4 → 4 = 2×2

So:
- Left top: 12
- Left middle: 4
- Right top: 3

✔ Completed:
```
36
/ \
12 3
/ \
3 4
/ \
2 2
```

---

#### l. ?
```
?
/ \
? ?
/ \ / \
? 2 ? 2
/ \ / \
2 3 2 2
```
- Left side: 2×3 = 6 → then 6×2 = 12 → so left top is 12
- Right side: 2×2 = 4 → then 4×2 = 8 → so right top is 8
- Then top: 12 × 8 = 96

So top number is 96

✔ Answer: Top is 96

---

Part 2: Use a factor tree to find the product of prime factors

#### a. 48
Let’s do a factor tree:
- 48 = 16 × 3
- 16 = 8 × 2
- 8 = 4 × 2
- 4 = 2 × 2
- So prime factors: 2, 2, 2, 2, 2, 3 → five 2s and one 3

Or: 48 = 2 × 2 × 2 × 2 × 2 × 3

So: 2 × 2 × 2 × 2 × 2 × 3

But usually written as: 2 × 2 × 2 × 2 × 2 × 3

So fill in:
48: 2 × 2 × 2 × 2 × 2 × 3

#### b. 100
- 100 = 10 × 10
- 10 = 2 × 5
- So 100 = 2 × 5 × 2 × 5 = 2 × 2 × 5 × 5

So: 2 × 2 × 5 × 5

#### c. 144
- 144 = 12 × 12
- 12 = 3 × 4 = 3 × 2 × 2
- So 144 = (3 × 2 × 2) × (3 × 2 × 2) = 2 × 2 × 2 × 2 × 3 × 3

So: 2 × 2 × 2 × 2 × 3 × 3

---

✔ Final Answers

---

Part 1: Prime Factor Trees

a.
```
20
/ \
4 5
/ \
2 2
```

b.
```
12
/ \
2 6
/ \
2 3
```

c.
```
24
/ \
4 6
/ \ / \
2 2 2 3
```

d.
```
40
/ \
8 5
/ \
4 2
/ \
2 2
```

e.
```
56
/ \
2 28
/ \
4 7
/ \
2 2
```

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

g. Top number: 16

h. Top number: 30

i.
```
75
/ \
25 3
/ \
5 5
```

j.
```
80
/ \
20 40
/ \ / \
4 5 4 10
/ \ / \
2 2 2 5
```
(But simplified: 80 → 20 and 40; 20 → 4 and 5; 4 → 2 and 2; 40 → 4 and 10; 4 → 2 and 2; 10 → 2 and 5)

k.
```
36
/ \
12 3
/ \
3 4
/ \
2 2
```

l. Top number: 96

---

Part 2: Product of Prime Factors

a. 48: 2 × 2 × 2 × 2 × 2 × 3

b. 100: 2 × 2 × 5 × 5

c. 144: 2 × 2 × 2 × 2 × 3 × 3

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Let me know if you'd like these formatted as a completed worksheet!