Find the HCF and LCM of `12, 15, 18, 27`

Options

A. 3 , 540
B. 3 , 270
C. 3, 54
D. none of these

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 12, 15, 18, and 27, we will follow these steps:

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

- **For 12**:
- \(12 = 2^2 \times 3^1\)

- **For 15**:
- \(15 = 3^1 \times 5^1\)

- **For 18**:
- \(18 = 2^1 \times 3^2\)

- **For 27**:
- \(27 = 3^3\)

### Step 2: Finding HCF
To find the HCF, we take the lowest power of all prime factors that appear in all factorizations.

- **Common prime factors**: The only common prime factor is \(3\).
- **Lowest power of 3**: \(3^1\)

Thus, the HCF is:
\[ \text{HCF} = 3^1 = 3 \]

### Step 3: Finding LCM
To find the LCM, we take the highest power of all prime factors that appear in any of the factorizations.

- **For 2**: The highest power is \(2^2\) (from 12).
- **For 3**: The highest power is \(3^3\) (from 27).
- **For 5**: The highest power is \(5^1\) (from 15).

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^3 \times 5^1 \]

Calculating this:
\[ = 4 \times 27 \times 5 \]
\[ = 108 \times 5 = 540 \]

### Final Results
- **HCF**: 3
- **LCM**: 540

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