Using factorisation, find the HCF of 36 and 54.

Answer

To find the Highest Common Factor (HCF) of 36 and 54 using the prime factorization method, we can follow these steps:

### Step 1: Prime Factorization of 36
We start by finding the prime factors of 36.

- Divide 36 by the smallest prime number, which is 2:
\[ 36 \div 2 = 18 \]
- Divide 18 by 2 again:
\[ 18 \div 2 = 9 \]
- Now, divide 9 by the next smallest prime number, which is 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

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

### Step 2: Prime Factorization of 54
Next, we find the prime factors of 54.

- Divide 54 by 2:
\[ 54 \div 2 = 27 \]
- Now, divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

Thus, 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 of 36 and 54.

- The prime factorization of 36 is \(2^2 \times 3^2\).
- The prime factorization of 54 is \(2^1 \times 3^3\).

The common prime factors are:
- For 2: The minimum power is \(2^1\).
- For 3: The minimum power is \(3^2\).

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

### Final Answer
The HCF of 36 and 54 is:
\[ \text{HCF} = 18 \]

---