Find the least common multiple of each pair of numbers on this math worksheet.

To solve the problem of finding the least common multiple (LCM) for each pair of numbers, we will use the prime factorization method. Here's a step-by-step explanation and solution for each pair:

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Step 1: Understand the LCM

The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. To find the LCM using prime factorization:
1. Find the prime factorization of each number.
2. For each prime factor, take the highest power that appears in the factorizations.
3. Multiply these highest powers together to get the LCM.

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Step 2: Solve Each Pair

#### (1) \( 15 \) and \( 20 \)

- Prime Factorization:
- \( 15 = 3 \times 5 \)
- \( 20 = 2^2 \times 5 \)

- Identify the Highest Powers:
- For \( 2 \): The highest power is \( 2^2 \) (from 20).
- For \( 3 \): The highest power is \( 3^1 \) (from 15).
- For \( 5 \): The highest power is \( 5^1 \) (common in both).

- Calculate the LCM:
\[
\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
\]

- Answer:
\[
\boxed{60}
\]

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#### (2) \( 8 \) and \( 12 \)

- Prime Factorization:
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3 \)

- Identify the Highest Powers:
- For \( 2 \): The highest power is \( 2^3 \) (from 8).
- For \( 3 \): The highest power is \( 3^1 \) (from 12).

- Calculate the LCM:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
\]

- Answer:
\[
\boxed{24}
\]

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#### (3) \( 9 \) and \( 15 \)

- Prime Factorization:
- \( 9 = 3^2 \)
- \( 15 = 3 \times 5 \)

- Identify the Highest Powers:
- For \( 3 \): The highest power is \( 3^2 \) (from 9).
- For \( 5 \): The highest power is \( 5^1 \) (from 15).

- Calculate the LCM:
\[
\text{LCM} = 3^2 \times 5^1 = 9 \times 5 = 45
\]

- Answer:
\[
\boxed{45}
\]

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#### (4) \( 18 \) and \( 27 \)

- Prime Factorization:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)

- Identify the Highest Powers:
- For \( 2 \): The highest power is \( 2^1 \) (from 18).
- For \( 3 \): The highest power is \( 3^3 \) (from 27).

- Calculate the LCM:
\[
\text{LCM} = 2^1 \times 3^3 = 2 \times 27 = 54
\]

- Answer:
\[
\boxed{54}
\]

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#### (5) \( 14 \) and \( 21 \)

- Prime Factorization:
- \( 14 = 2 \times 7 \)
- \( 21 = 3 \times 7 \)

- Identify the Highest Powers:
- For \( 2 \): The highest power is \( 2^1 \) (from 14).
- For \( 3 \): The highest power is \( 3^1 \) (from 21).
- For \( 7 \): The highest power is \( 7^1 \) (common in both).

- Calculate the LCM:
\[
\text{LCM} = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42
\]

- Answer:
\[
\boxed{42}
\]

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#### (6) \( 24 \) and \( 36 \)

- Prime Factorization:
- \( 24 = 2^3 \times 3 \)
- \( 36 = 2^2 \times 3^2 \)

- Identify the Highest Powers:
- For \( 2 \): The highest power is \( 2^3 \) (from 24).
- For \( 3 \): The highest power is \( 3^2 \) (from 36).

- Calculate the LCM:
\[
\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72
\]

- Answer:
\[
\boxed{72}
\]

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Final Answers:

\[
\boxed{60, 24, 45, 54, 42, 72}
\]