Find the LCM of the numbers given below :
144, 180, 384

Answer

To find the LCM (Least Common Multiple) of the numbers 144, 180, and 384, we will follow these steps:

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

- **For 144**:
- 144 = 12 × 12
- 12 = 3 × 4 = 3 × 2 × 2
- Therefore, 144 = 3 × 2 × 2 × 3 × 2 × 2 = \(2^4 × 3^2\)

- **For 180**:
- 180 = 18 × 10
- 18 = 2 × 9 = 2 × 3 × 3
- 10 = 2 × 5
- Therefore, 180 = 2 × 3 × 3 × 2 × 5 = \(2^2 × 3^2 × 5^1\)

- **For 384**:
- 384 = 192 × 2
- 192 = 96 × 2
- 96 = 48 × 2
- 48 = 24 × 2
- 24 = 12 × 2
- 12 = 6 × 2
- 6 = 3 × 2
- Therefore, 384 = \(2^7 × 3^1\)

### Step 2: Identify the Highest Powers of Each Prime Factor
Next, we will identify the highest powers of all prime factors present in the factorizations:

- For the prime factor **2**:
- Highest power is \(2^7\) (from 384)

- For the prime factor **3**:
- Highest power is \(3^2\) (from both 144 and 180)

- For the prime factor **5**:
- Highest power is \(5^1\) (from 180)

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors:

\[ \text{LCM} = 2^7 × 3^2 × 5^1 \]

Calculating this step-by-step:

1. Calculate \(2^7 = 128\)
2. Calculate \(3^2 = 9\)
3. Calculate \(5^1 = 5\)

Now multiply these results together:

\[ \text{LCM} = 128 × 9 × 5 \]

Calculating \(128 × 9 = 1152\)

Now, multiply \(1152 × 5 = 5760\)

### Final Answer
Thus, the LCM of 144, 180, and 384 is **5760**.

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