The HCF of 36 and 54 is :

Options

A. 2
B. 6
C. 9
D. 18

Answer

To find the HCF (Highest Common Factor) of 36 and 54, we can use the method of prime factorization. Here’s the step-by-step solution:

### Step 1: Prime Factorization of 36
1. Start dividing 36 by the smallest prime number, which is 2.
- \( 36 \div 2 = 18 \)
2. Divide 18 by 2 again.
- \( 18 \div 2 = 9 \)
3. Now, divide 9 by the next smallest prime number, which is 3.
- \( 9 \div 3 = 3 \)
4. Finally, divide 3 by 3.
- \( 3 \div 3 = 1 \)

So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

### Step 2: Prime Factorization of 54
1. Start dividing 54 by 2.
- \( 54 \div 2 = 27 \)
2. Now, divide 27 by the next smallest prime number, which is 3.
- \( 27 \div 3 = 9 \)
3. Divide 9 by 3 again.
- \( 9 \div 3 = 3 \)
4. Finally, divide 3 by 3.
- \( 3 \div 3 = 1 \)

So, the prime factorization of 54 is:
\[ 54 = 2^1 \times 3^3 \]

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:
- For 36: \( 2^2 \times 3^2 \)
- For 54: \( 2^1 \times 3^3 \)

The common prime factors are:
- For the prime number 2: the minimum exponent is \( 1 \) (from 54).
- For the prime number 3: the minimum exponent is \( 2 \) (from 36).

### Step 4: Calculate the HCF
Now, we can calculate the HCF by multiplying the common factors:
\[ \text{HCF} = 2^{\text{min}(2,1)} \times 3^{\text{min}(2,3)} \]
\[ \text{HCF} = 2^1 \times 3^2 \]
\[ \text{HCF} = 2 \times 9 = 18 \]

Thus, the HCF of 36 and 54 is **18**.

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