To identify the least common multiple (LCM) of the given sets of numbers, we follow these steps:

1. Prime factorization of each number.
2. Identify the highest power of each prime number involved.
3. Multiply these highest powers together to get the LCM.

Part (a): 22, 33, and 44

Prime factorization:

• $22 = 2 \times 11$
• $33 = 3 \times 11$
• $44 = 2^2 \times 11$

Highest powers of prime factors:

• $2^2$
• $3$
• $11$

LCM calculation: $\text{LCM} = 2^2 \times 3 \times 11 = 4 \times 3 \times 11 = 132$

Part (b): 15 and 20

Prime factorization:

• $15 = 3 \times 5$
• $20 = 2^2 \times 5$

Highest powers of prime factors:

• $2^2$
• $3$
• $5$

LCM calculation: $\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$

Part (c): 10, 15, and 25

Prime factorization:

• $10 = 2 \times 5$
• $15 = 3 \times 5$
• $25 = 5^2$

Highest powers of prime factors:

• $2$
• $3$
• $5^2$

LCM calculation: $\text{LCM} = 2 \times 3 \times 5^2 = 2 \times 3 \times 25 = 150$

Final Results

• (a) LCM of 22, 33, and 44: 132
• (b) LCM of 15 and 20: 60
• (c) LCM of 10, 15, and 25: 150

Would you like details on any part of the calculation or have any other questions?

Here are some questions you might ask next:

1. How do you find the greatest common divisor (GCD) of a set of numbers?
2. Can you explain the prime factorization method in more detail?
3. What is the difference between LCM and GCD?
4. How do you solve linear equations?
5. What are the properties of prime numbers?

Tip: When finding the LCM, always break down the numbers into their prime factors to ensure accuracy in your calculations.