The HCF of 156, 312 and 195 is

Options

A. 39
B. 3
C. 78
D. 13

Answer

To find the HCF (Highest Common Factor) of the numbers 156, 312, and 195, we will use the prime factorization method. Here are the steps:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 156:**
- Divide by 2: \( 156 \div 2 = 78 \)
- Divide by 2: \( 78 \div 2 = 39 \)
- Divide by 3: \( 39 \div 3 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 156 is:
\[ 156 = 2^2 \times 3^1 \times 13^1 \]

2. **Prime Factorization of 312:**
- Divide by 2: \( 312 \div 2 = 156 \)
- Divide by 2: \( 156 \div 2 = 78 \)
- Divide by 2: \( 78 \div 2 = 39 \)
- Divide by 3: \( 39 \div 3 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 312 is:
\[ 312 = 2^3 \times 3^1 \times 13^1 \]

3. **Prime Factorization of 195:**
- Divide by 3: \( 195 \div 3 = 65 \)
- Divide by 5: \( 65 \div 5 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 195 is:
\[ 195 = 3^1 \times 5^1 \times 13^1 \]

### Step 2: Identify the Common Factors

Now, we will identify the common prime factors from the factorizations obtained:

- From 156: \( 2^2 \times 3^1 \times 13^1 \)
- From 312: \( 2^3 \times 3^1 \times 13^1 \)
- From 195: \( 3^1 \times 5^1 \times 13^1 \)

### Step 3: Determine the HCF

To find the HCF, we take the lowest power of all common prime factors:

- For \( 2 \): The minimum power is \( 0 \) (since 195 has no factor of 2).
- For \( 3 \): The minimum power is \( 1 \).
- For \( 13 \): The minimum power is \( 1 \).

Thus, the HCF is calculated as:
\[ HCF = 2^0 \times 3^1 \times 13^1 = 1 \times 3 \times 13 = 39 \]

### Final Answer
The HCF of 156, 312, and 195 is **39**.

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