Solved 4. GCF & LCM Using Venn Diagrams Work with a partner ... - Free Printable

Let's solve both problems step by step using Venn diagrams to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM).

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🔷 Problem 1: Find GCF and LCM of 72 and 54

#### Step 1: Prime Factorization

- 72
$ 72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2 \times 2 \times 2 \times 3 \times 3 $
So, $ 72 = 2^3 \times 3^2 $

- 54
$ 54 = 2 \times 27 = 2 \times 3 \times 9 = 2 \times 3 \times 3 \times 3 $
So, $ 54 = 2^1 \times 3^3 $

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#### Step 2: Place prime factors in the Venn Diagram

We'll draw two overlapping circles:
- One for 72
- One for 54
- The intersection contains common prime factors (to the lowest power)

##### Common primes:
- Both have 2 → lowest power is $ 2^1 $
- Both have 3 → lowest power is $ 3^2 $ (since 72 has $ 3^2 $, 54 has $ 3^3 $; take minimum)

So, place:
- In intersection: $ 2 $ and $ 3 $, $ 3 $ → so write 2, 3, 3 in the middle

##### Unique to 72:
- Extra $ 2^2 $ → two more 2s → place 2, 2 in the 72 circle (outside intersection)

##### Unique to 54:
- Extra $ 3^1 $ → one more 3 → place 3 in the 54 circle (outside intersection)

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#### Venn Diagram Layout:

```
72 ∩ 54
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ 2 │ │ 2 │ │ 3 │
│ 2 │ │ 3 │ │ │
│ │ │ 3 │ │ │
└─────────────┘ └─────────────┘ └─────────────┘
```

Wait — better layout:

Actually, we should represent it like this:

```
72 ∩ 54
┌───────────────┐ ┌───────────────┐ ┌───────────────┐
│ 2 │ │ 2 │ │ 3 │
│ 2 │ │ 3 │ │ │
│ │ │ 3 │ │ │
└───────────────┘ └───────────────┘ └───────────────┘
```

But since the intersection holds common factors, let’s clarify:

- Intersection: $ 2 \times 3 \times 3 = 2 \times 3^2 $
- Only 72: extra $ 2 \times 2 = 2^2 $
- Only 54: extra $ 3 = 3^1 $

So the diagram should be:

```
72 ∩ 54
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ 2 │ │ 2 │ │ 3 │
│ 2 │ │ 3 │ │ │
│ │ │ 3 │ │ │
└─────────────┘ └─────────────┘ └─────────────┘
```

But actually, since the intersection includes all shared primes to the lowest power, we write:

- Intersection: 2, 3, 3
- Left (only 72): 2, 2
- Right (only 54): 3

So final Venn diagram:

```
72 ∩ 54
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ 2 │ │ 2 │ │ 3 │
│ 2 │ │ 3 │ │ │
│ │ │ 3 │ │ │
└─────────────┘ └─────────────┘ └─────────────┘
```

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#### Step 3: Find GCF and LCM

- GCF = product of numbers in the intersection
$ 2 \times 3 \times 3 = 18 $

- LCM = product of all numbers in the diagram
$ 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3 = 8 \times 27 = 216 $

✔ So:
- GCF of 72 and 54 = 18
- LCM of 72 and 54 = 216

---

🔷 Problem 2: Find GCF and LCM of 27 and 36

#### Step 1: Prime Factorization

- 27
$ 27 = 3 \times 9 = 3 \times 3 \times 3 = 3^3 $

- 36
$ 36 = 6 \times 6 = (2 \times 3) \times (2 \times 3) = 2^2 \times 3^2 $

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#### Step 2: Place in Venn Diagram

- Common primes: only 3 → lowest power is $ 3^2 $ (since 27 has $ 3^3 $, 36 has $ 3^2 $)
- So intersection: $ 3 $, $ 3 $

- Only in 27: extra $ 3^1 $ → one more 3
- Only in 36: $ 2^2 $ → two 2s

So:
- Intersection: 3, 3
- Only 27: 3
- Only 36: 2, 2

Venn Diagram:

```
27 ∩ 36
┌─────────────┐ ┌─────────────┐ ┌─────────────┐
│ 3 │ │ 3 │ │ 2 │
│ │ │ 3 │ │ 2 │
│ │ │ │ │ │
└─────────────┘ └─────────────┘ └─────────────┘
```

---

#### Step 3: Find GCF and LCM

- GCF = product of intersection = $ 3 \times 3 = 9 $
- LCM = product of all numbers = $ 3 \times 3 \times 3 \times 2 \times 2 = 3^3 \times 2^2 = 27 \times 4 = 108 $

✔ So:
- GCF of 27 and 36 = 9
- LCM of 27 and 36 = 108

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✔ Final Answers:

#### 1) 72 and 54
- GCF = 18
- LCM = 216

#### 2) 27 and 36
- GCF = 9
- LCM = 108

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📝 Summary of Method:

- Prime factorize both numbers.
- Place shared primes (lowest power) in the intersection.
- Unique primes go in their respective circles.
- GCF = product of intersection
- LCM = product of all numbers in the diagram (union)

Let me know if you'd like a visual sketch!