Multiples and Factors worksheet with questions on identifying multiples, finding least common multiples, and listing factors.

Let's solve each problem step by step.

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Problem 1: Write two multiples of 5 between 15 and 30.

Solution:
- Multiples of 5 are numbers that can be written as \( 5 \times n \), where \( n \) is an integer.
- We need multiples of 5 between 15 and 30.
- The multiples of 5 in this range are: 20, 25.
- Therefore, two multiples of 5 between 15 and 30 are 20 and 25.

Answer:
\[
\boxed{20, 25}
\]

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Problem 2: Circle the numbers that are multiples of 8.

Given numbers:
\[ 8, \, 16, \, 20, \, 24, \, 30, \, 32, \, 36, \, 40 \]

Solution:
- A multiple of 8 is a number that can be written as \( 8 \times n \), where \( n \) is an integer.
- Check each number:
- \( 8 = 8 \times 1 \) (multiple of 8)
- \( 16 = 8 \times 2 \) (multiple of 8)
- \( 20 \neq 8 \times n \) (not a multiple of 8)
- \( 24 = 8 \times 3 \) (multiple of 8)
- \( 30 \neq 8 \times n \) (not a multiple of 8)
- \( 32 = 8 \times 4 \) (multiple of 8)
- \( 36 \neq 8 \times n \) (not a multiple of 8)
- \( 40 = 8 \times 5 \) (multiple of 8)

- The multiples of 8 are: \( 8, 16, 24, 32, 40 \).

Answer:
Circle:
\[
\boxed{8, 16, 24, 32, 40}
\]

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Problem 3: Write the first three multiples of:

#### a) 13
#### b) 14
#### c) 17

Solution:
- The first three multiples of a number \( n \) are \( n \times 1 \), \( n \times 2 \), and \( n \times 3 \).

#### a) First three multiples of 13:
\[ 13 \times 1 = 13, \, 13 \times 2 = 26, \, 13 \times 3 = 39 \]
- The first three multiples of 13 are: 13, 26, 39.

#### b) First three multiples of 14:
\[ 14 \times 1 = 14, \, 14 \times 2 = 28, \, 14 \times 3 = 42 \]
- The first three multiples of 14 are: 14, 28, 42.

#### c) First three multiples of 17:
\[ 17 \times 1 = 17, \, 17 \times 2 = 34, \, 17 \times 3 = 51 \]
- The first three multiples of 17 are: 17, 34, 51.

Answer:
\[
\boxed{13, 26, 39 \quad 14, 28, 42 \quad 17, 34, 51}
\]

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Problem 4: Write the least common multiple (LCM) of 2, 8, and 12.

Solution:
- To find the LCM, we use the prime factorization method.
- Prime factorizations:
- \( 2 = 2 \)
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3 \)
- The LCM is found by taking the highest power of each prime factor:
- For \( 2 \): The highest power is \( 2^3 \) (from 8).
- For \( 3 \): The highest power is \( 3^1 \) (from 12).
- Therefore, the LCM is:
\[ \text{LCM} = 2^3 \times 3 = 8 \times 3 = 24 \]

Answer:
\[
\boxed{24}
\]

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Problem 5: Write the least common multiple for each set of numbers.

#### a) 12 and 20
#### b) 6 and 14
#### c) 11 and 15

Solution:

#### a) LCM of 12 and 20:
- Prime factorizations:
- \( 12 = 2^2 \times 3 \)
- \( 20 = 2^2 \times 5 \)
- Highest powers of each prime factor:
- For \( 2 \): \( 2^2 \)
- For \( 3 \): \( 3^1 \)
- For \( 5 \): \( 5^1 \)
- Therefore, the LCM is:
\[ \text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \]

#### b) LCM of 6 and 14:
- Prime factorizations:
- \( 6 = 2 \times 3 \)
- \( 14 = 2 \times 7 \)
- Highest powers of each prime factor:
- For \( 2 \): \( 2^1 \)
- For \( 3 \): \( 3^1 \)
- For \( 7 \): \( 7^1 \)
- Therefore, the LCM is:
\[ \text{LCM} = 2 \times 3 \times 7 = 42 \]

#### c) LCM of 11 and 15:
- Prime factorizations:
- \( 11 = 11 \)
- \( 15 = 3 \times 5 \)
- Highest powers of each prime factor:
- For \( 11 \): \( 11^1 \)
- For \( 3 \): \( 3^1 \)
- For \( 5 \): \( 5^1 \)
- Therefore, the LCM is:
\[ \text{LCM} = 11 \times 3 \times 5 = 165 \]

Answer:
\[
\boxed{60, 42, 165}
\]

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Problem 6: List all the factors for each number. Is the number prime?

#### a) 87
#### b) 196

Solution:

#### a) Factors of 87:
- To find the factors, check divisibility by integers from 1 to \( \sqrt{87} \approx 9.3 \).
- Divisibility checks:
- \( 87 \div 1 = 87 \) → Factors: \( 1, 87 \)
- \( 87 \div 3 = 29 \) → Factors: \( 3, 29 \)
- No other divisors between 4 and 9.
- The factors of 87 are: 1, 3, 29, 87.
- Since 87 has more than two factors (1 and itself), it is not prime.

#### b) Factors of 196:
- To find the factors, check divisibility by integers from 1 to \( \sqrt{196} = 14 \).
- Divisibility checks:
- \( 196 \div 1 = 196 \) → Factors: \( 1, 196 \)
- \( 196 \div 2 = 98 \) → Factors: \( 2, 98 \)
- \( 196 \div 4 = 49 \) → Factors: \( 4, 49 \)
- \( 196 \div 7 = 28 \) → Factors: \( 7, 28 \)
- No other divisors between 8 and 13.
- The factors of 196 are: 1, 2, 4, 7, 14, 28, 49, 98, 196.
- Since 196 has more than two factors (1 and itself), it is not prime.

Answer:
\[
\boxed{1, 3, 29, 87 \quad \text{(Not prime)} \quad 1, 2, 4, 7, 14, 28, 49, 98, 196 \quad \text{(Not prime)}}
\]

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Final Answers:

1. \(\boxed{20, 25}\)
2. \(\boxed{8, 16, 24, 32, 40}\)
3. \(\boxed{13, 26, 39 \quad 14, 28, 42 \quad 17, 34, 51}\)
4. \(\boxed{24}\)
5. \(\boxed{60, 42, 165}\)
6. \(\boxed{1, 3, 29, 87 \quad \text{(Not prime)} \quad 1, 2, 4, 7, 14, 28, 49, 98, 196 \quad \text{(Not prime)}}\)