Math worksheet for finding factors and multiples using Venn diagrams and a table.

Let's solve each part of the worksheet step by step.

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Problem 1: Venn Diagram – Multiples of 5 and Multiples of 30

We are given a list of numbers:

Numbers:
1, 12, 15, 10, 2, 5, 35, 3, 8

We need to place them in the correct sections of the Venn diagram:
- Left circle: Multiples of 5
- Right circle: Multiples of 30
- Intersection: Multiples of both 5 and 30, i.e., Multiples of 30 (since all multiples of 30 are also multiples of 5)

Let’s check which numbers from the list are multiples of 5 or 30.

#### Step-by-step analysis:

| Number | Multiple of 5? | Multiple of 30? |
|--------|----------------|------------------|
| 1 | No | No |
| 12 | No | No |
| 15 | Yes | No |
| 10 | Yes | No |
| 2 | No | No |
| 5 | Yes | No |
| 35 | Yes | No |
| 3 | No | No |
| 8 | No | No |

Now, note:
- Multiples of 30 must be divisible by 30 → 30, 60, 90, etc.
- None of the numbers listed are multiples of 30.

So:
- Only multiples of 5 (but not 30): 15, 10, 5, 35
- Multiples of 30: none (so intersection is empty)
- Numbers that are neither: 1, 12, 2, 3, 8

But wait — we’re only supposed to use the numbers given alongside. So we only include numbers from the list.

✔ So for Venn Diagram 1:

- Left circle (only multiples of 5): 15, 10, 5, 35
- Right circle (multiples of 30): none (no number in the list is a multiple of 30)
- Intersection: none (since no number is a multiple of 30)

> But wait — 30 itself isn’t in the list. So even though 30 is a multiple of both 5 and 30, it's not among the options.

So final answer for Problem 1:

- Multiples of 5 only: 5, 10, 15, 35
- Multiples of 30 only: none
- Both: none

But let’s double-check: Is 30 a multiple of 5? Yes. Is it a multiple of 30? Yes. But 30 is not in the list, so we don’t include it.

So we just place the numbers from the list that are multiples of 5 in the left circle.

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✔ Final Answer for Problem 1:

- Left Circle (Multiples of 5 only): 5, 10, 15, 35
- Right Circle (Multiples of 30 only): none
- Intersection: none

> Since there are no multiples of 30 in the list, the right circle and intersection remain blank.

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Problem 2: Venn Diagram – Multiples of 3 and Multiples of 7

Given numbers:

List:
12, 21, 20, 14, 25, 35, 18, 30, 42

We need to classify them into:
- Multiples of 3
- Multiples of 7
- Both (i.e., multiples of LCM(3,7)=21)

Let’s check each number:

| Number | Divisible by 3? | Divisible by 7? | Category |
|--------|------------------|------------------|----------|
| 12 | Yes (12 ÷ 3 = 4) | No | Only 3 |
| 21 | Yes (21 ÷ 3 = 7) | Yes (21 ÷ 7 = 3)| Both |
| 20 | No | No | Neither |
| 14 | No | Yes (14 ÷ 7 = 2)| Only 7 |
| 25 | No | No | Neither |
| 35 | No | Yes (35 ÷ 7 = 5)| Only 7 |
| 18 | Yes (18 ÷ 3 = 6) | No | Only 3 |
| 30 | Yes (30 ÷ 3 = 10)| No | Only 3 |
| 42 | Yes (42 ÷ 3 = 14)| Yes (42 ÷ 7 = 6)| Both |

Now group:

- Only Multiples of 3: 12, 18, 30
- Only Multiples of 7: 14, 35
- Both (Multiples of 3 and 7): 21, 42
- Neither: 20, 25

✔ So for Venn Diagram 2:

- Left Circle (Multiples of 3 only): 12, 18, 30
- Right Circle (Multiples of 7 only): 14, 35
- Intersection (Multiples of both 3 and 7): 21, 42

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✔ Final Answer for Problem 2:

- Multiples of 3 only: 12, 18, 30
- Multiples of 7 only: 14, 35
- Both: 21, 42

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Problem 3: Fill the table with 'Yes' or 'No'

We are to determine:
- Is the number a multiple of 3?
- Is it a multiple of 36?

Let’s go one by one.

| Number | Multiple of 3? | Multiple of 36? |
|--------|----------------|------------------|
| 4 | No | Yes (Wait — is 4 a multiple of 36?) ✘ Let's recheck |

Wait! The example says:
- 4 → Multiple of 3? No → Correct
- Multiple of 36? Yes → ✘ This is incorrect

Because 4 is NOT a multiple of 36. A multiple of 36 must be ≥36 or 0, and divisible by 36.

But 4 < 36, and 36 doesn't divide 4. So "Yes" is wrong.

But the problem says: "One is done for you." And it shows:
> Number: 4 → Multiple of 3? No → Multiple of 36? Yes

This is wrong.

But maybe there's a typo? Or perhaps they meant "Is 36 a multiple of 4?" but no — it says “Multiple of 36?”

Let’s assume the example is correct, so maybe they made a mistake.

But logically:
- A multiple of 36 means: 36, 72, 108, ...
- 4 is not a multiple of 36.
- So the example is wrong.

But since the problem says “one is done for you,” and gives:
> 4 → No, Yes

We must follow the logic, so perhaps the question is asking something else?

Wait — maybe it's asking: "Is the number a multiple of 36?" — meaning does 36 divide the number?

So:
- Is 4 a multiple of 36? → No (because 36 × 0 = 0, 36 × 1 = 36 > 4)
- So should be "No"

But the table says "Yes" — this is an error.

But since it's given as an example, perhaps it's a typo. Let's proceed carefully.

Alternatively, could it be that the number 36 is being tested? But no — the row is for number 4.

So I think the example is incorrect. But we’ll continue with correct math.

Let’s fill the table correctly.

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#### Check each number:

1. 4
- Multiple of 3? → 4 ÷ 3 = 1.33… → No
- Multiple of 36? → 36 × 0 = 0, 36 × 1 = 36 → 4 < 36 → No
- But table says Yes → ✘ Incorrect

2. 6
- 6 ÷ 3 = 2 → Yes
- 6 < 36 → not a multiple of 36 → No

3. 10
- 10 ÷ 3 ≈ 3.33 → No
- 10 < 36 → No

4. 12
- 12 ÷ 3 = 4 → Yes
- 12 < 36 → No

5. 15
- 15 ÷ 3 = 5 → Yes
- 15 < 36 → No

6. 21
- 21 ÷ 3 = 7 → Yes
- 21 < 36 → No

So corrected table:

| Number | Multiple of 3? | Multiple of 36? |
|--------|----------------|------------------|
| 4 | No | No (not Yes!) |
| 6 | Yes | No |
| 10 | No | No |
| 12 | Yes | No |
| 15 | Yes | No |
| 21 | Yes | No |

But the example says 4 → Yes for multiple of 36 — which is wrong.

Possibility: Maybe the column is reversed?

Wait — maybe the question is: “Is 36 a multiple of the number?” But that would be different.

For example:
- Is 36 a multiple of 4? → Yes, because 36 ÷ 4 = 9 → Yes
- Is 36 a multiple of 6? → Yes, 36 ÷ 6 = 6
- Is 36 a multiple of 10? → No
- Is 36 a multiple of 12? → Yes (36 ÷ 12 = 3)
- Is 36 a multiple of 15? → 36 ÷ 15 = 2.4 → No
- Is 36 a multiple of 21? → 36 ÷ 21 ≈ 1.71 → No

Ah! That makes sense.

So maybe the column "Multiple of 36?" is actually asking: "Is 36 a multiple of this number?"

That is, does the number divide 36?

Let’s test that.

Check if 36 is divisible by the number.

So:
- 4: 36 ÷ 4 = 9 → Yes → matches the example ✔
- 6: 36 ÷ 6 = 6 → Yes
- 10: 36 ÷ 10 = 3.6 → No
- 12: 36 ÷ 12 = 3 → Yes
- 15: 36 ÷ 15 = 2.4 → No
- 21: 36 ÷ 21 ≈ 1.71 → No

And for multiple of 3:
- 4 → not divisible by 3 → No
- 6 → yes
- 10 → no
- 12 → yes
- 15 → yes
- 21 → yes

So now the table becomes:

| Number | Multiple of 3? | Multiple of 36? (i.e., Does 36 ÷ number = integer?) |
|--------|----------------|------------------------------------------------------|
| 4 | No | Yes (36 ÷ 4 = 9) |
| 6 | Yes | Yes (36 ÷ 6 = 6) |
| 10 | No | No (36 ÷ 10 = 3.6) |
| 12 | Yes | Yes (36 ÷ 12 = 3) |
| 15 | Yes | No (36 ÷ 15 = 2.4) |
| 21 | Yes | No (36 ÷ 21 ≈ 1.71) |

This matches the example perfectly.

So the column "Multiple of 36?" means: "Is 36 a multiple of this number?", i.e., is this number a factor of 36?

That makes sense.

So the correct interpretation is:
- "Multiple of 3?" → Is the number divisible by 3?
- "Multiple of 36?" → Is 36 divisible by the number? i.e., is the number a factor of 36?

✔ That explains why 4 → Yes (since 4 divides 36), even though 4 is not a multiple of 36.

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✔ Final Table:

| Number | Multiple of 3? | Multiple of 36? (i.e., factor of 36?) |
|--------|----------------|---------------------------------------|
| 4 | No | Yes |
| 6 | Yes | Yes |
| 10 | No | No |
| 12 | Yes | Yes |
| 15 | Yes | No |
| 21 | Yes | No |

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✔ Summary of Answers:

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Problem 1: Venn Diagram – Multiples of 5 and 30

- Multiples of 5 only: 5, 10, 15, 35
- Multiples of 30 only: none
- Both: none

> So place: 5, 10, 15, 35 in left circle; nothing in right or intersection.

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Problem 2: Venn Diagram – Multiples of 3 and 7

- Multiples of 3 only: 12, 18, 30
- Multiples of 7 only: 14, 35
- Both: 21, 42

> Place accordingly.

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Problem 3: Table

| Number | Multiple of 3? | Multiple of 36? |
|--------|----------------|------------------|
| 4 | No | Yes |
| 6 | Yes | Yes |
| 10 | No | No |
| 12 | Yes | Yes |
| 15 | Yes | No |
| 21 | Yes | No |

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✔ All problems solved with correct reasoning.

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