Find the HCF of 52 and 320:

Options

A. 16
B. 13
C. 4
D. 18

Answer

To find the Highest Common Factor (HCF) or the Greatest Common Divisor (GCD) of 52 and 320, we can use the prime factorization method. Here’s the step-by-step solution:

### Step 1: Prime Factorization of 52
- Start by dividing 52 by the smallest prime number, which is 2.
- \( 52 \div 2 = 26 \)
- Divide 26 by 2 again.
- \( 26 \div 2 = 13 \)
- Now, 13 is a prime number and cannot be divided further.

**Prime factorization of 52:**
\[ 52 = 2^2 \times 13^1 \]

### Step 2: Prime Factorization of 320
- Start by dividing 320 by the smallest prime number, which is 2.
- \( 320 \div 2 = 160 \)
- Divide 160 by 2.
- \( 160 \div 2 = 80 \)
- Divide 80 by 2.
- \( 80 \div 2 = 40 \)
- Divide 40 by 2.
- \( 40 \div 2 = 20 \)
- Divide 20 by 2.
- \( 20 \div 2 = 10 \)
- Divide 10 by 2.
- \( 10 \div 2 = 5 \)
- Now, 5 is a prime number and cannot be divided further.

**Prime factorization of 320:**
\[ 320 = 2^6 \times 5^1 \]

### Step 3: Identify Common Factors
- Now, we compare the prime factors of both numbers.
- For 52: \( 2^2 \times 13^1 \)
- For 320: \( 2^6 \times 5^1 \)

The common prime factor is 2.

### Step 4: Find the Lowest Power of Common Factors
- The lowest power of the common factor (which is 2) in both factorizations is \( 2^2 \).

### Step 5: Calculate the HCF
- The HCF is the product of the common prime factors raised to their lowest powers.
\[ \text{HCF} = 2^2 = 4 \]

### Final Answer
The HCF of 52 and 320 is **4**.

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