Find the LCM of 12, 16 and 18 by prime factorisation method.

Options

A. 112
B. 120
C. 144
D. 150

Answer

To find the LCM of 12, 16, and 18 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number
1. **Factorizing 12:**
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 12 is:
\[ 12 = 2^2 \times 3^1 \]

2. **Factorizing 16:**
- 16 can be divided by 2: \( 16 \div 2 = 8 \)
- 8 can be divided by 2: \( 8 \div 2 = 4 \)
- 4 can be divided by 2: \( 4 \div 2 = 2 \)
- 2 is a prime number.
- Therefore, the prime factorization of 16 is:
\[ 16 = 2^4 \]

3. **Factorizing 18:**
- 18 can be divided by 2: \( 18 \div 2 = 9 \)
- 9 can be divided by 3: \( 9 \div 3 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 18 is:
\[ 18 = 2^1 \times 3^2 \]

### Step 2: Identify the Highest Powers of Each Prime Factor
Now we will list all the prime factors and take the highest power of each:
- For the prime factor 2:
- From 12: \( 2^2 \)
- From 16: \( 2^4 \)
- From 18: \( 2^1 \)
- Highest power is \( 2^4 \).

- For the prime factor 3:
- From 12: \( 3^1 \)
- From 16: \( 3^0 \) (not present)
- From 18: \( 3^2 \)
- Highest power is \( 3^2 \).

### Step 3: Calculate the LCM
Now, we multiply the highest powers of all prime factors:
\[ \text{LCM} = 2^4 \times 3^2 \]

Calculating this:
- \( 2^4 = 16 \)
- \( 3^2 = 9 \)
- Therefore,
\[ \text{LCM} = 16 \times 9 = 144 \]

### Final Answer
The LCM of 12, 16, and 18 is **144**.

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