To find the Least Common Multiple (LCM) of 5, 15, 30, and 40, follow these steps:

### Step 1: Prime Factorization
- \( 5 = 5 \)
- \( 15 = 3 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- \( 40 = 2^3 \times 5 \)

### Step 2: Identify the highest powers of all primes
The LCM is determined by taking the highest power of each prime factor:
- For \( 2 \), the highest power is \( 2^3 \) (from 40).
- For \( 3 \), the highest power is \( 3 \) (from 15 and 30).
- For \( 5 \), the highest power is \( 5 \) (from all numbers).

### Step 3: Multiply the highest powers of each prime
\[
\text{LCM} = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120
\]

### Conclusion:
The LCM of 5, 15, 30, and 40 is **120**.

Would you like further clarification or details on any part of this process?

Here are some related questions to expand on:
1. What is the LCM of 12, 18, and 24?
2. How is the LCM different from the greatest common divisor (GCD)?
3. Can the LCM of two numbers ever be smaller than one of the numbers?
4. How can you use the LCM to solve problems involving fractions?
5. How does prime factorization help in finding the LCM?

**Tip:** The LCM is useful when adding or comparing fractions with different denominators!

Learn how to find the Least Common Multiple (LCM) of 5, 15, 30, and 40 using prime factorization. This detailed solution explains each step and covers the prime factorization theorem, suitable for students in Grades 6-8.

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