Find the LCM of 24, 96 and 36 .

Options

A. 576
B. 216
C. 288
D. 144

Answer

To find the LCM (Least Common Multiple) of the numbers 24, 96, and 36, we can follow these steps:

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

- **24**:
- Divide by 2: 24 ÷ 2 = 12
- Divide by 2: 12 ÷ 2 = 6
- Divide by 2: 6 ÷ 2 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 24 is \( 2^3 \times 3^1 \).

- **96**:
- Divide by 2: 96 ÷ 2 = 48
- Divide by 2: 48 ÷ 2 = 24
- Divide by 2: 24 ÷ 2 = 12
- Divide by 2: 12 ÷ 2 = 6
- Divide by 2: 6 ÷ 2 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 96 is \( 2^5 \times 3^1 \).

- **36**:
- Divide by 2: 36 ÷ 2 = 18
- Divide by 2: 18 ÷ 2 = 9
- Divide by 3: 9 ÷ 3 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 36 is \( 2^2 \times 3^2 \).

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

- For \( 2 \): The highest power is \( 2^5 \) (from 96).
- For \( 3 \): The highest power is \( 3^2 \) (from 36).

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

\[ LCM = 2^5 \times 3^2 \]

Calculating this gives:

\[ 2^5 = 32 \]
\[ 3^2 = 9 \]
\[ LCM = 32 \times 9 = 288 \]

### Final Answer
Thus, the LCM of 24, 96, and 36 is **288**.

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