In the following questions, each question has four options (A), (B), (C) and (D). Choose the correct option and indicate your correct response.
The H.C.F. of 104, 169, 117 is:

Options

A. 17
B. 13
C. 19
D. 23

Answer

To find the H.C.F. (Highest Common Factor) of the numbers 104, 169, and 117, we will follow these steps:

### Step 1: Factor each number into its prime factors.

1. **Factor 104:**
- 104 is even, so we can divide by 2.
- 104 ÷ 2 = 52
- 52 is also even, so we divide by 2 again.
- 52 ÷ 2 = 26
- 26 is even, so we divide by 2 once more.
- 26 ÷ 2 = 13
- 13 is a prime number.
- Therefore, the prime factorization of 104 is:
\[ 104 = 2^3 \times 13 \]

2. **Factor 169:**
- 169 is not even, so we check for divisibility by 3, 5, etc.
- We find that 169 = 13 × 13.
- Therefore, the prime factorization of 169 is:
\[ 169 = 13^2 \]

3. **Factor 117:**
- 117 is not even, so we check for divisibility by 3.
- 117 ÷ 3 = 39
- 39 is also divisible by 3.
- 39 ÷ 3 = 13
- Therefore, the prime factorization of 117 is:
\[ 117 = 3^2 \times 13 \]

### Step 2: Identify the common prime factors.

From the prime factorizations:
- 104 = \(2^3 \times 13\)
- 169 = \(13^2\)
- 117 = \(3^2 \times 13\)

The only common prime factor among these numbers is **13**.

### Step 3: Determine the H.C.F.

Since 13 is the only common prime factor, the H.C.F. of 104, 169, and 117 is:
\[ \text{H.C.F.} = 13 \]

### Conclusion

The H.C.F. of 104, 169, and 117 is **13**.