Math worksheet for practicing factors and multiples using Venn diagrams.

Let's solve each of the four Venn diagram problems step by step, explaining how to determine where each number goes.

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Problem 1: Factors of 48 and Factors of 32

We are given numbers:
1, 12, 16, 3, 2, 24, 48, 32, 4, 8, 6

We need to place them in the correct regions of a Venn diagram:
- Left circle: Factors of 48
- Right circle: Factors of 32
- Intersection: Common factors (factors of both 48 and 32)

#### Step 1: Find all factors
- Factors of 48:
1, 2, 3, 4, 6, 8, 12, 16, 24, 48

- Factors of 32:
1, 2, 4, 8, 16, 32

#### Step 2: Identify common factors (intersection)
- Common factors: 1, 2, 4, 8, 16

#### Step 3: Place numbers
Now go through the list:

| Number | Factor of 48? | Factor of 32? | Position |
|--------|----------------|----------------|---------|
| 1 | Yes | Yes | ✔ Intersection |
| 12 | Yes | No | Left only |
| 16 | Yes | Yes | ✔ Intersection |
| 3 | Yes | No | Left only |
| 2 | Yes | Yes | ✔ Intersection |
| 24 | Yes | No | Left only |
| 48 | Yes | No | Left only |
| 32 | No | Yes | Right only |
| 4 | Yes | Yes | ✔ Intersection |
| 8 | Yes | Yes | ✔ Intersection |
| 6 | Yes | No | Left only |

#### Final Placement:
- Left only (only factors of 48): 12, 3, 24, 48, 6
- Right only (only factors of 32): 32
- Intersection (common factors): 1, 2, 4, 8, 16

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Problem 2: Multiples of 2 and Multiples of 5

Numbers:
10, 5, 20, 50, 30, 12, 48, 2

Venn diagram:
- Left: Multiples of 2
- Right: Multiples of 5
- Intersection: Multiples of both 2 and 5 → i.e., multiples of 10

#### Step 1: Check each number

| Number | Multiple of 2? | Multiple of 5? | Position |
|--------|----------------|----------------|---------|
| 10 | Yes | Yes | ✔ Intersection |
| 5 | No | Yes | Right only |
| 20 | Yes | Yes | ✔ Intersection |
| 50 | Yes | Yes | ✔ Intersection |
| 30 | Yes | Yes | ✔ Intersection |
| 12 | Yes | No | Left only |
| 48 | Yes | No | Left only |
| 2 | Yes | No | Left only |

#### Final Placement:
- Left only (multiples of 2 but not 5): 12, 48, 2
- Right only (multiples of 5 but not 2): 5
- Intersection (multiples of both 2 and 5): 10, 20, 50, 30

> Note: All multiples of 10 are multiples of both 2 and 5.

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Problem 3: Factors of 36 and Factors of 40

Numbers:
5, 10, 9, 12, 18, 40, 20, 8, 1, 4, 2, 36, 3, 6

#### Step 1: Find factors

- Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36

- Factors of 40:
1, 2, 4, 5, 8, 10, 20, 40

#### Step 2: Common factors (intersection)
- Common: 1, 2, 4

#### Step 3: Place each number

| Number | Factor of 36? | Factor of 40? | Position |
|--------|----------------|----------------|---------|
| 5 | No | Yes | Right only |
| 10 | No | Yes | Right only |
| 9 | Yes | No | Left only |
| 12 | Yes | No | Left only |
| 18 | Yes | No | Left only |
| 40 | No | Yes | Right only |
| 20 | No | Yes | Right only |
| 8 | No | Yes | Right only |
| 1 | Yes | Yes | ✔ Intersection |
| 4 | Yes | Yes | ✔ Intersection |
| 2 | Yes | Yes | ✔ Intersection |
| 36 | Yes | No | Left only |
| 3 | Yes | No | Left only |
| 6 | Yes | No | Left only |

#### Final Placement:
- Left only (factors of 36 only): 9, 12, 18, 36, 3, 6
- Right only (factors of 40 only): 5, 10, 40, 20, 8
- Intersection (common factors): 1, 2, 4

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Problem 4: Multiples of 4 and Multiples of 6

Numbers:
4, 6, 18, 20, 32, 44, 30, 42, 12, 16, 28, 8, 36, 48, 8, 40

Note: "8" appears twice — we’ll treat it once.

We want:
- Left: Multiples of 4
- Right: Multiples of 6
- Intersection: Multiples of both 4 and 6 → LCM(4,6) = 12 → so multiples of 12

#### Step 1: Check each number

| Number | Multiple of 4? | Multiple of 6? | Position |
|--------|----------------|----------------|---------|
| 4 | Yes | No | Left only |
| 6 | No | Yes | Right only |
| 18 | No | Yes | Right only |
| 20 | Yes | No | Left only |
| 32 | Yes | No | Left only |
| 44 | Yes | No | Left only |
| 30 | No | Yes | Right only |
| 42 | No | Yes | Right only |
| 12 | Yes | Yes | ✔ Intersection |
| 16 | Yes | No | Left only |
| 28 | Yes | No | Left only |
| 8 | Yes | No | Left only |
| 36 | Yes | Yes | ✔ Intersection |
| 48 | Yes | Yes | ✔ Intersection |
| 40 | Yes | No | Left only |

Wait — let’s double-check:

- Multiples of 4: divisible by 4 → last two digits divisible by 4.
- Multiples of 6: divisible by both 2 and 3 → even and sum of digits divisible by 3.

Check:
- 12: 12 ÷ 4 = 3 ✔️, 12 ÷ 6 = 2 ✔️ → ✔ intersection
- 36: 36 ÷ 4 = 9 ✔️, 36 ÷ 6 = 6 ✔️ → ✔ intersection
- 48: 48 ÷ 4 = 12 ✔️, 48 ÷ 6 = 8 ✔️ → ✔ intersection

Any others?

- 24 is not in list.
- 72 not here.

So only 12, 36, 48 are in intersection.

Now confirm:

| Number | Multiple of 4? | Multiple of 6? | Position |
|--------|----------------|----------------|---------|
| 4 | Yes | No | Left only |
| 6 | No | Yes | Right only |
| 18 | No | Yes | Right only |
| 20 | Yes | No | Left only |
| 32 | Yes | No | Left only |
| 44 | Yes | No | Left only |
| 30 | No | Yes | Right only |
| 42 | No | Yes | Right only |
| 12 | Yes | Yes | ✔ Intersection |
| 16 | Yes | No | Left only |
| 28 | Yes | No | Left only |
| 8 | Yes | No | Left only |
| 36 | Yes | Yes | ✔ Intersection |
| 48 | Yes | Yes | ✔ Intersection |
| 40 | Yes | No | Left only |

#### Final Placement:
- Left only (multiples of 4 only): 4, 20, 32, 44, 16, 28, 8, 40
- Right only (multiples of 6 only): 6, 18, 30, 42
- Intersection (multiples of both 4 and 6): 12, 36, 48

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✔ Summary of Answers:

#### 1. Factors of 48 and 32
- Only 48: 12, 3, 24, 48, 6
- Only 32: 32
- Both: 1, 2, 4, 8, 16

#### 2. Multiples of 2 and 5
- Only 2: 12, 48, 2
- Only 5: 5
- Both: 10, 20, 50, 30

#### 3. Factors of 36 and 40
- Only 36: 9, 12, 18, 36, 3, 6
- Only 40: 5, 10, 40, 20, 8
- Both: 1, 2, 4

#### 4. Multiples of 4 and 6
- Only 4: 4, 20, 32, 44, 16, 28, 8, 40
- Only 6: 6, 18, 30, 42
- Both: 12, 36, 48

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You can now fill in the Venn diagrams accordingly. Let me know if you'd like a visual layout!