To find the **LCM (Least Common Multiple)** and **GCF (Greatest Common Factor)** of 32 and 64, let's break it down step by step.

### 1. Finding the GCF (Greatest Common Factor):
The GCF is the largest number that divides both 32 and 64 without leaving a remainder.

- The prime factorization of 32: \( 32 = 2^5 \)
- The prime factorization of 64: \( 64 = 2^6 \)

The common factors are powers of 2, and we take the lowest exponent in both factorizations. So, the GCF is:
\[
GCF(32, 64) = 2^5 = 32
\]

### 2. Finding the LCM (Least Common Multiple):
The LCM is the smallest number that both 32 and 64 divide into without leaving a remainder.

Using their prime factorizations:
- \( 32 = 2^5 \)
- \( 64 = 2^6 \)

For the LCM, we take the highest power of each prime factor. So, the LCM is:
\[
LCM(32, 64) = 2^6 = 64
\]

### Final Answer:
- **GCF(32, 64) = 32**
- **LCM(32, 64) = 64**

Would you like more details or have any further questions?

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Here are 5 related questions to expand on the topic:
1. How do we find the GCF of more than two numbers?
2. What is the LCM of 12, 15, and 20?
3. How does prime factorization help in finding GCF and LCM?
4. Can the LCM ever be smaller than the GCF?
5. How are LCM and GCF used in solving real-world problems?

**Tip:** The GCF is always less than or equal to the smallest of the numbers, while the LCM is always greater than or equal to the largest number!

Learn how to find the LCM (Least Common Multiple) and GCF (Greatest Common Factor) of 32 and 64 using prime factorization. This detailed solution explains step-by-step methods, including breaking down the prime factorizations of 32 and 64, to determine that GCF(32, 64) = 32 and LCM(32, 64) = 64. Ideal for students in Grades 6-8.

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