Identify the LCM of the numbers 32, 16 and 9 among the options.

Options

A. 9
B. 32
C. 114
D. 228

Answer

To find the LCM (Least Common Multiple) of the numbers 32, 16, and 9, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **32**:
- 32 = 2 × 16
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
- So, the prime factorization of 32 is \(2^5\).

- **16**:
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
- So, the prime factorization of 16 is \(2^4\).

- **9**:
- 9 = 3 × 3
- So, the prime factorization of 9 is \(3^2\).

### Step 2: Identify the Highest Powers
Next, we identify the highest power of each prime number from the factorizations:

- For the prime number **2**:
- The highest power is \(2^5\) (from 32).

- For the prime number **3**:
- The highest power is \(3^2\) (from 9).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2^5 \times 3^2 \]

Calculating this step-by-step:

- \(2^5 = 32\)
- \(3^2 = 9\)

Now multiply these results:

\[ \text{LCM} = 32 \times 9 = 288 \]

### Conclusion
The LCM of 32, 16, and 9 is **288**.