The LCM of 20, 24, 30 and 32 would be

Options

A. 398
B. 410
C. 450
D. 480

Answer

To find the LCM (Least Common Multiple) of the numbers 20, 24, 30, and 32, we will use the prime factorization method. Here are the steps:

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

- **20**:
- 20 = 2 × 10 = 2 × 2 × 5 = \(2^2 \times 5^1\)

- **24**:
- 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = \(2^3 \times 3^1\)

- **30**:
- 30 = 2 × 15 = 2 × 3 × 5 = \(2^1 \times 3^1 \times 5^1\)

- **32**:
- 32 = 2 × 16 = 2 × 2 × 8 = 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 = \(2^5\)

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

- For **2**: The highest power is \(2^5\) (from 32).
- For **3**: The highest power is \(3^1\) (from 24 and 30).
- For **5**: The highest power is \(5^1\) (from 20 and 30).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:

\[ \text{LCM} = 2^5 \times 3^1 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \(2^5 = 32\)
2. Calculate \(3^1 = 3\)
3. Calculate \(5^1 = 5\)

Now, multiply these results together:

\[ \text{LCM} = 32 \times 3 \times 5 \]

Calculating \(32 \times 3 = 96\)

Now multiply \(96 \times 5\):

\[ 96 \times 5 = 480 \]

### Final Answer
Thus, the LCM of 20, 24, 30, and 32 is **480**.

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