Find the HCF of 48, 64 and 78

Options

A. `8`
B. `10`
C. `12`
D. `6`

Answer

To find the Highest Common Factor (HCF) of the numbers 48, 64, and 78 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number
1. **Prime Factorization of 48**:
- Divide 48 by 2: \(48 \div 2 = 24\)
- Divide 24 by 2: \(24 \div 2 = 12\)
- Divide 12 by 2: \(12 \div 2 = 6\)
- Divide 6 by 2: \(6 \div 2 = 3\)
- Divide 3 by 3: \(3 \div 3 = 1\)
- So, the prime factorization of 48 is \(2^4 \times 3^1\).

2. **Prime Factorization of 64**:
- Divide 64 by 2: \(64 \div 2 = 32\)
- Divide 32 by 2: \(32 \div 2 = 16\)
- Divide 16 by 2: \(16 \div 2 = 8\)
- Divide 8 by 2: \(8 \div 2 = 4\)
- Divide 4 by 2: \(4 \div 2 = 2\)
- Divide 2 by 2: \(2 \div 2 = 1\)
- So, the prime factorization of 64 is \(2^6\).

3. **Prime Factorization of 78**:
- Divide 78 by 2: \(78 \div 2 = 39\)
- Divide 39 by 3: \(39 \div 3 = 13\)
- Divide 13 by 13: \(13 \div 13 = 1\)
- So, the prime factorization of 78 is \(2^1 \times 3^1 \times 13^1\).

### Step 2: Identify Common Prime Factors
Now, we list the prime factors obtained from the factorization:
- 48: \(2^4 \times 3^1\)
- 64: \(2^6\)
- 78: \(2^1 \times 3^1 \times 13^1\)

### Step 3: Find Common Factors
The common prime factor among all three numbers is \(2\).
- The minimum power of \(2\) in the factorizations is \(2^1\) (from 78).

### Step 4: Calculate the HCF
Thus, the HCF of 48, 64, and 78 is:
\[ HCF = 2^1 = 2 \]

### Final Answer
The HCF of 48, 64, and 78 is **2**.

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