Factors And Multiples - Math Steps, Examples & Questions - Free Printable

Problem Analysis and Solution

The worksheet consists of three main tasks. Let's solve each one step by step.

---

Task 1: Fill in the table below

The table has three columns:
1. NUMBER: A list of numbers.
2. FACTOR OF 15?: Determine if the number is a factor of 15.
3. MULTIPLES OF 3?: Determine if the number is a multiple of 3.

#### Step-by-Step Solution:

1. Factors of 15:
- The factors of 15 are the numbers that divide 15 evenly. They are: \(1, 3, 5, 15\).

2. Multiples of 3:
- A number is a multiple of 3 if it can be expressed as \(3 \times k\) for some integer \(k\). For example, \(3, 6, 9, 12, 15, \ldots\).

Now, let's fill in the table for each number:

- Number: 3
- Factor of 15?: Yes (since \(3\) divides \(15\) evenly).
- Multiple of 3?: Yes (since \(3 = 3 \times 1\)).

- Number: 30
- Factor of 15?: No (since \(30\) does not divide \(15\) evenly).
- Multiple of 3?: Yes (since \(30 = 3 \times 10\)).

- Number: 6
- Factor of 15?: No (since \(6\) does not divide \(15\) evenly).
- Multiple of 3?: Yes (since \(6 = 3 \times 2\)).

- Number: 12
- Factor of 15?: No (since \(12\) does not divide \(15\) evenly).
- Multiple of 3?: Yes (since \(12 = 3 \times 4\)).

- Number: 20
- Factor of 15?: No (since \(20\) does not divide \(15\) evenly).
- Multiple of 3?: No (since \(20\) is not divisible by \(3\)).

#### Final Table:
| NUMBER | FACTOR OF 15? | MULTIPLES OF 3? |
|--------|----------------|------------------|
| 3 | Yes | Yes |
| 30 | No | Yes |
| 6 | No | Yes |
| 12 | No | Yes |
| 20 | No | No |

---

Task 2: Complete the factor tree

#### Factor Tree for 60:
1. Start with 60.
2. Break it down into two factors: \(60 = 30 \times 2\).
3. Break down 30 further: \(30 = 10 \times 3\).
4. Break down 10 further: \(10 = 5 \times 2\).

So, the complete factor tree for 60 is:
```
60
/ \
30 2
/ \
10 3
/ \
5 2
```

#### Factor Tree for 48:
1. Start with 48.
2. Break it down into two factors: \(48 = 4 \times 12\).
3. Break down 4 further: \(4 = 2 \times 2\).
4. Break down 12 further: \(12 = 3 \times 4\).
5. Break down 4 again: \(4 = 2 \times 2\).

So, the complete factor tree for 48 is:
```
48
/ \
4 12
/ \ / \
2 2 3 4
/ \
2 2
```

#### Final Factor Trees:
- For 60:
```
60
/ \
30 2
/ \
10 3
/ \
5 2
```

- For 48:
```
48
/ \
4 12
/ \ / \
2 2 3 4
/ \
2 2
```

---

Task 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?

#### Step-by-Step Solution:

1. Identify multiples of 60 between 400 and 600:
- The multiples of 60 are: \(60 \times 7 = 420\), \(60 \times 8 = 480\), \(60 \times 9 = 540\), \(60 \times 10 = 600\).
- Between 400 and 600, the multiples are: \(420, 480, 540\).

2. Check which of these is 40 away from a multiple of 100:
- 420: Closest multiples of 100 are 400 and 500.
- Distance from 400: \(420 - 400 = 20\)
- Distance from 500: \(500 - 420 = 80\)
- Not 40 away from a multiple of 100.
- 480: Closest multiples of 100 are 400 and 500.
- Distance from 400: \(480 - 400 = 80\)
- Distance from 500: \(500 - 480 = 20\)
- Not 40 away from a multiple of 100.
- 540: Closest multiples of 100 are 500 and 600.
- Distance from 500: \(540 - 500 = 40\)
- Distance from 600: \(600 - 540 = 60\)
- This is 40 away from 500.

3. Conclusion:
- The number that satisfies both conditions is \(540\).

#### Final Answer:
\[
\boxed{540}
\]

---

Final Summary of Answers:

1. Table Filling:
| NUMBER | FACTOR OF 15? | MULTIPLES OF 3? |
|--------|----------------|------------------|
| 3 | Yes | Yes |
| 30 | No | Yes |
| 6 | No | Yes |
| 12 | No | Yes |
| 20 | No | No |

2. Factor Trees:
- For 60:
```
60
/ \
30 2
/ \
10 3
/ \
5 2
```
- For 48:
```
48
/ \
4 12
/ \ / \
2 2 3 4
/ \
2 2
```

3. Who am I?:
\[
\boxed{540}
\]