Find the LCM of 24, 36 and 40 by the prime factorization method.

Options

A. 360
B. 420
C. 300
D. 280

Answer

To find the LCM (Least Common Multiple) of the numbers 24, 36, and 40 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 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 \( 2^3 \times 3^1 \).

- **For 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 \( 2^2 \times 3^2 \).

- **For 40**:
- Divide by 2: 40 ÷ 2 = 20
- Divide by 2: 20 ÷ 2 = 10
- Divide by 2: 10 ÷ 2 = 5
- Divide by 5: 5 ÷ 5 = 1
- So, the prime factorization of 40 is \( 2^3 \times 5^1 \).

### Step 2: Identify the Highest Powers of Each Prime Factor
Now, we will list the prime factors and take the highest power of each:
- For the prime factor **2**: The highest power is \( 2^3 \) (from both 24 and 40).
- For the prime factor **3**: The highest power is \( 3^2 \) (from 36).
- For the prime factor **5**: The highest power is \( 5^1 \) (from 40).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \]
Calculating this step-by-step:
- \( 2^3 = 8 \)
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)

Now multiply them:
\[ 8 \times 9 = 72 \]
Then,
\[ 72 \times 5 = 360 \]

### Final Answer
Thus, the LCM of 24, 36, and 40 is **360**.

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