Find the appropriate relation for quantity 1 and quantity 2 in the following question :
Quantity I : the unit digit in `(6817)^(754)`
Quantity II : the unit digit in `(3^(65)xx6^(59)xx7^(71))`

Options

A. A)Quantity I `gt` Quantity II
B. B)Quantity I `lt` Quantity II
C. C)Quantity I `ge` Quantity II
D. D)Quantity I `le` Quantity II

Answer

To solve the problem, we need to find the unit digits of two quantities separately and then compare them.

### Step 1: Find the unit digit of Quantity I
**Quantity I**: The unit digit in \( (6817)^{754} \)

1. Identify the unit digit of the base number \( 6817 \). The unit digit is \( 7 \).
2. We need to find the unit digit of \( 7^{754} \). The unit digits of powers of \( 7 \) follow a pattern:
- \( 7^1 \) = 7 (unit digit is 7)
- \( 7^2 \) = 49 (unit digit is 9)
- \( 7^3 \) = 343 (unit digit is 3)
- \( 7^4 \) = 2401 (unit digit is 1)
- The pattern repeats every 4 terms: 7, 9, 3, 1.
3. To find the relevant term in the pattern, calculate \( 754 \mod 4 \):
- \( 754 \div 4 = 188 \) remainder \( 2 \).
4. The remainder \( 2 \) corresponds to the second term in the pattern, which is \( 9 \).

Thus, the unit digit of \( (6817)^{754} \) is **9**.

### Step 2: Find the unit digit of Quantity II
**Quantity II**: The unit digit in \( (3^{65} \times 6^{59} \times 7^{71}) \)

1. **Calculate the unit digit of \( 3^{65} \)**:
- The unit digits of powers of \( 3 \) follow a pattern:
- \( 3^1 \) = 3
- \( 3^2 \) = 9
- \( 3^3 \) = 27 (unit digit is 7)
- \( 3^4 \) = 81 (unit digit is 1)
- The pattern repeats every 4 terms: 3, 9, 7, 1.
- Calculate \( 65 \mod 4 \):
- \( 65 \div 4 = 16 \) remainder \( 1 \).
- The remainder \( 1 \) corresponds to the first term in the pattern, which is \( 3 \).

2. **Calculate the unit digit of \( 6^{59} \)**:
- The unit digit of any power of \( 6 \) is always \( 6 \).

3. **Calculate the unit digit of \( 7^{71} \)**:
- The unit digits of powers of \( 7 \) follow the same pattern as before: 7, 9, 3, 1.
- Calculate \( 71 \mod 4 \):
- \( 71 \div 4 = 17 \) remainder \( 3 \).
- The remainder \( 3 \) corresponds to the third term in the pattern, which is \( 3 \).

4. **Combine the unit digits**:
- Now we have:
- Unit digit of \( 3^{65} \) = \( 3 \)
- Unit digit of \( 6^{59} \) = \( 6 \)
- Unit digit of \( 7^{71} \) = \( 3 \)
- Multiply the unit digits together:
- \( 3 \times 6 = 18 \) (unit digit is \( 8 \))
- \( 8 \times 3 = 24 \) (unit digit is \( 4 \))

Thus, the unit digit of \( (3^{65} \times 6^{59} \times 7^{71}) \) is **4**.

### Step 3: Compare Quantity I and Quantity II
- Quantity I = 9
- Quantity II = 4

Since \( 9 > 4 \), we conclude that:

**Quantity I is greater than Quantity II.**

### Final Answer
**The appropriate relation is: Quantity I > Quantity II.**