Factors and Multiples Worksheets - Math Monks - Free Printable

Let's solve each part of the Factors and Multiples Worksheet step by step.

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1) Fill in the table below.

We are given a table with numbers, and we need to determine:
- Whether the number is a factor of 15
- Whether it is a multiple of 3

| NUMBER | FACTOR OF 15? | MULTIPLE OF 3? |
|--------|----------------|----------------|
| 3 | | |
| 30 | | |
| 6 | | |
| 12 | | |
| 20 | | |

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#### Step 1: Understand what "factor of 15" means
A factor of 15 is a number that divides 15 exactly (no remainder).

The factors of 15 are:
1, 3, 5, 15

So any number in the list that is one of these is a factor of 15.

#### Step 2: Understand what "multiple of 3" means
A multiple of 3 is a number that can be divided by 3 with no remainder.
In other words, if a number ÷ 3 has no remainder, it’s a multiple of 3.

Let’s check each number:

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Number: 3
- Is 3 a factor of 15? → Yes (3 × 5 = 15)
- Is 3 a multiple of 3? → Yes (3 ÷ 3 = 1)

✔ Factor of 15: Yes
✔ Multiple of 3: Yes

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Number: 30
- Is 30 a factor of 15? → No, because 15 ÷ 30 = 0.5 → not a whole number.
- Is 30 a multiple of 3? → Yes (30 ÷ 3 = 10)

✘ Factor of 15: No
✔ Multiple of 3: Yes

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Number: 6
- Is 6 a factor of 15? → 15 ÷ 6 = 2.5 → No
- Is 6 a multiple of 3? → 6 ÷ 3 = 2 → Yes

✘ Factor of 15: No
✔ Multiple of 3: Yes

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Number: 12
- Is 12 a factor of 15? → 15 ÷ 12 = 1.25 → No
- Is 12 a multiple of 3? → 12 ÷ 3 = 4 → Yes

✘ Factor of 15: No
✔ Multiple of 3: Yes

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Number: 20
- Is 20 a factor of 15? → 15 ÷ 20 = 0.75 → No
- Is 20 a multiple of 3? → 20 ÷ 3 = 6.666... → No

✘ Factor of 15: No
✘ Multiple of 3: No

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✔ Completed Table:

| NUMBER | FACTOR OF 15? | MULTIPLE OF 3? |
|--------|----------------|----------------|
| 3 | Yes | Yes |
| 30 | No | Yes |
| 6 | No | Yes |
| 12 | No | Yes |
| 20 | No | No |

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2) Complete the factor tree.

We have two factor trees: one for 60, one for 48.

We break down each number into prime factors using factor trees.

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#### Factor Tree for 60

```
60
/ \
30 ?
/ \
? ?
/ \
? ?
```

We know:
- 60 = 30 × 2 → So the right branch is 2
- 30 = 10 × 3 → So left is 10, right is 3
- 10 = 5 × 2 → So left is 5, right is 2

Let’s fill it in:

```
60
/ \
30 2
/ \
10 3
/ \
5 2
```

✔ Final factorization: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

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#### Factor Tree for 48

```
48
/ \
? 24
/ \
? ?
/ \
? ?
/ \
? ?
```

We know:
- 48 = 2 × 24 → So left branch is 2
- 24 = 8 × 3 → So left is 8, right is 3
- 8 = 4 × 2 → So left is 4, right is 2
- 4 = 2 × 2 → So left is 2, right is 2

Now fill in:

```
48
/ \
2 24
/ \
8 3
/ \
4 2
/ \
2 2
```

✔ Final factorization: 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

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✔ Completed Factor Trees:

60:
```
60
/ \
30 2
/ \
10 3
/ \
5 2
```

48:
```
48
/ \
2 24
/ \
8 3
/ \
4 2
/ \
2 2
```

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3) I am a multiple of 60. I am between 400 and 600. I am 40 away from a multiple of 100. Who am I?

Let’s break this down.

We are looking for a number that satisfies:

1. Multiple of 60
2. Between 400 and 600
3. 40 away from a multiple of 100

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#### Step 1: List multiples of 60 between 400 and 600

Multiples of 60:
- 60, 120, 180, 240, 300, 360, 420, 480, 540, 600

But between 400 and 600 → so exclude 600 (since it's not *between*)

So possible candidates: 420, 480, 540

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#### Step 2: Which of these are 40 away from a multiple of 100?

Multiples of 100 near this range: 400, 500, 600

Check distance from each candidate:

- 420:
- Distance to 400 = 20 → not 40
- Distance to 500 = 80 → not 40
✘

- 480:
- To 400 = 80
- To 500 = 20
✘

- 540:
- To 500 = 40 → YES!
- To 600 = 60 → not 40
✔

So 540 is 40 away from 500, which is a multiple of 100.

Also:
- 540 ÷ 60 = 9 → yes, it's a multiple of 60
- 400 < 540 < 600 → yes

✔ All conditions satisfied.

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✔ Answer: 540

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✔ Final Answers:

#### 1) Table

| NUMBER | FACTOR OF 15? | MULTIPLE OF 3? |
|--------|----------------|----------------|
| 3 | Yes | Yes |
| 30 | No | Yes |
| 6 | No | Yes |
| 12 | No | Yes |
| 20 | No | No |

#### 2) Factor Trees

60:
```
60
/ \
30 2
/ \
10 3
/ \
5 2
```

48:
```
48
/ \
2 24
/ \
8 3
/ \
4 2
/ \
2 2
```

#### 3) The number is: 540

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✔ All problems solved!