What is the HCF of 40 and 54 ?

Answer

To find the Highest Common Factor (HCF) of 40 and 54, we can follow these steps:

### Step 1: Prime Factorization of 40
To find the prime factors of 40, we can divide it by the smallest prime number until we reach 1.

- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 40 is:
\[ 40 = 2^3 \times 5^1 \]

### Step 2: Prime Factorization of 54
Now, we will find the prime factors of 54 in a similar manner.

- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

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

### Step 3: Identify Common Factors
Next, we need to identify the common prime factors from both numbers.

From the prime factorizations:
- For 40: \( 2^3 \times 5^1 \)
- For 54: \( 2^1 \times 3^3 \)

The only common prime factor is \( 2 \).

### Step 4: Determine the HCF
To find the HCF, we take the lowest power of the common prime factor:
- The common prime factor is \( 2 \).
- The lowest power of \( 2 \) in both factorizations is \( 2^1 \).

Thus, the HCF of 40 and 54 is:
\[ HCF = 2^1 = 2 \]

### Final Answer
The HCF of 40 and 54 is **2**.

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