Find the LCM of 30, 45, 63 by prime factorization

To find the Least Common Multiple (LCM) of the numbers 30, 45, and 63 using prime factorization, we follow these steps:

1. **Find the prime factors of each number:**

   - **30:**
     \[
     30 = 2 \times 3 \times 5
     \]

   - **45:**
     \[
     45 = 3^2 \times 5
     \]

   - **63:**
     \[
     63 = 3^2 \times 7
     \]

2. **Identify the highest powers of all prime numbers that appear in these factorizations:**
   - The prime numbers involved are 2, 3, 5, and 7.
   - The highest power of 2 in the factorizations is \(2^1\).
   - The highest power of 3 in the factorizations is \(3^2\).
   - The highest power of 5 in the factorizations is \(5^1\).
   - The highest power of 7 in the factorizations is \(7^1\).

3. **Multiply these highest powers together to find the LCM:**
   \[
   \text{LCM} = 2^1 \times 3^2 \times 5^1 \times 7^1
   \]

4. **Calculate the product:**
   \[
   2^1 = 2
   \]
   \[
   3^2 = 9
   \]
   \[
   5^1 = 5
   \]
   \[
   7^1 = 7
   \]
   \[
   \text{LCM} = 2 \times 9 \times 5 \times 7
   \]
   \[
   \text{LCM} = 2 \times 9 = 18
   \]
   \[
   18 \times 5 = 90
   \]
   \[
   90 \times 7 = 630
   \]

So, the LCM of 30, 45, and 63 is \(630\).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the LCM of two numbers using the Euclidean algorithm?
2. What is the difference between LCM and GCD?
3. Can you explain the method to find the GCD of multiple numbers?
4. How do you find the LCM of algebraic expressions?
5. What is the significance of LCM in solving real-world problems?
6. How can you use LCM to solve problems involving time intervals?
7. What are some common mistakes when calculating LCM?
8. How do you find the LCM of decimal numbers?

**Tip:** Always break down the problem into smaller parts and solve step by step to avoid confusion and errors.

Learn how to find the Least Common Multiple (LCM) of 30, 45, and 63 using prime factorization. Step-by-step explanation and calculations included. Suitable for students in Grades 7-9 studying number theory.

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