To ** find the LCM of two numbers, 9 and 12**, we’ll use a method called

**prime factorization**. This involves breaking down each number into its prime factors (the prime numbers that multiply together to give the original number). Then, we’ll use those prime factorizations to determine the LCM.

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### Step 1: Prime Factorization of 9

First, let’s break down 9 into its prime factors:

- 9 can be written as 3 × 3.
- So, the prime factorization of 9 is
**3²**.

### Step 2: Prime Factorization of 12

Next, we factor 12:

- 12 can be written as 2 × 6. Then we factor 6: 6 = 2 × 3.
- So, the prime factorization of 12 is
**2² × 3**.

### Step 3: List the Prime Factors

- 9 = 3²
- 12 = 2² × 3

### Step 4: Take the Highest Powers of Each Prime Factor

For the LCM, we need to take the **highest powers** of all the prime factors that appear in either factorization.

- The highest power of 2 is
**2²**(from 12). - The highest power of 3 is
**3²**(from 9).

### Step 5: Multiply the Highest Powers Together

Now, we multiply the highest powers of each prime factor to get the LCM:

- LCM = 2² × 3²
- LCM = 4 × 9
- LCM =
**36**

### Final Answer:

The **LCM of 9 and 12** is **36**.

### Why this works:

The LCM is the smallest number that both 9 and 12 can divide into evenly. By taking the highest powers of the prime factors from both numbers, we ensure that all factors needed to represent both numbers are included in the LCM.