Quantity I: Unit digit of the number `[(4653)^(25)]^(27)`
Quantity II: Unit digit of the number `[(257)^(23)]^(22)`

Options

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

Explanation

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

### Step 1: Find the unit digit of Quantity I: \([(4653)^{25}]^{27}\)

1. **Identify the unit digit of the base**: The unit digit of 4653 is 3.
2. **Simplify the expression**: We need to find the unit digit of \(3^{(25 \times 27)}\).
3. **Calculate the exponent**:
\[ 25 \times 27 = 675 \]
4. **Find the unit digit of \(3^{675}\)**:
- The unit digits of powers of 3 follow a pattern:
- \(3^1 = 3\) (unit digit is 3)
- \(3^2 = 9\) (unit digit is 9)
- \(3^3 = 27\) (unit digit is 7)
- \(3^4 = 81\) (unit digit is 1)
- This pattern (3, 9, 7, 1) repeats every 4 terms.
5. **Determine the position in the cycle**:
- Calculate \(675 \mod 4\):
\[ 675 \div 4 = 168 \quad \text{(remainder 3)} \]
- The remainder is 3, which corresponds to the unit digit of \(3^3\), which is 7.
6. **Conclusion for Quantity I**: The unit digit of \([(4653)^{25}]^{27}\) is **7**.

### Step 2: Find the unit digit of Quantity II: \([(257)^{23}]^{22}\)

1. **Identify the unit digit of the base**: The unit digit of 257 is 7.
2. **Simplify the expression**: We need to find the unit digit of \(7^{(23 \times 22)}\).
3. **Calculate the exponent**:
\[ 23 \times 22 = 506 \]
4. **Find the unit digit of \(7^{506}\)**:
- 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)
- This pattern (7, 9, 3, 1) repeats every 4 terms.
5. **Determine the position in the cycle**:
- Calculate \(506 \mod 4\):
\[ 506 \div 4 = 126 \quad \text{(remainder 2)} \]
- The remainder is 2, which corresponds to the unit digit of \(7^2\), which is 9.
6. **Conclusion for Quantity II**: The unit digit of \([(257)^{23}]^{22}\) is **9**.

### Step 3: Compare the two quantities

- Quantity I: Unit digit is **7**.
- Quantity II: Unit digit is **9**.

### Final Conclusion:
Quantity I is smaller than Quantity II.