Find the LCM of 96, 144 and 150.

Options

A. 1140
B. 3600
C. 7200
D. 2400

Answer

To find the LCM (Least Common Multiple) of the numbers 96, 144, and 150, we will follow these steps:

### Step 1: Prime Factorization
We will start by finding the prime factorization of each number.

**For 96:**
1. Divide by 2: \( 96 \div 2 = 48 \)
2. Divide by 2: \( 48 \div 2 = 24 \)
3. Divide by 2: \( 24 \div 2 = 12 \)
4. Divide by 2: \( 12 \div 2 = 6 \)
5. Divide by 2: \( 6 \div 2 = 3 \)
6. Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

**For 144:**
1. Divide by 2: \( 144 \div 2 = 72 \)
2. Divide by 2: \( 72 \div 2 = 36 \)
3. Divide by 2: \( 36 \div 2 = 18 \)
4. Divide by 2: \( 18 \div 2 = 9 \)
5. Divide by 3: \( 9 \div 3 = 3 \)
6. Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

**For 150:**
1. Divide by 2: \( 150 \div 2 = 75 \)
2. Divide by 3: \( 75 \div 3 = 25 \)
3. Divide by 5: \( 25 \div 5 = 5 \)
4. Divide by 5: \( 5 \div 5 = 1 \)

So, the prime factorization of 150 is:
\[ 150 = 2^1 \times 3^1 \times 5^2 \]

### Step 2: Identify the Highest Powers of Each Prime Factor
Now, we will identify the highest power 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 144).
- For \(5\): The highest power is \(5^2\) (from 150).

### Step 3: Calculate the LCM
The LCM is found by multiplying the highest powers of all prime factors:
\[ \text{LCM} = 2^5 \times 3^2 \times 5^2 \]

Calculating this step-by-step:
1. Calculate \(2^5 = 32\)
2. Calculate \(3^2 = 9\)
3. Calculate \(5^2 = 25\)

Now multiply these results together:
\[ \text{LCM} = 32 \times 9 \times 25 \]

Calculating \(32 \times 9\):
\[ 32 \times 9 = 288 \]

Now calculate \(288 \times 25\):
\[ 288 \times 25 = 7200 \]

Thus, the LCM of 96, 144, and 150 is:
\[ \text{LCM} = 7200 \]

### Final Answer
The LCM of 96, 144, and 150 is **7200**.

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