The unit of `N^(2p)` and `(N)^((2p - 1))` are 6 and 4 respectively. The greatest two digit number that follows the afore - said condition is :

Options

A. 49
B. 84
C. 94
D. 98

Answer

To solve the problem, we need to find the greatest two-digit number \( N \) such that the unit digit of \( N^{2p} \) is 6 and the unit digit of \( N^{(2p - 1)} \) is 4.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find \( N \) such that:
- The unit digit of \( N^{2p} \) is 6.
- The unit digit of \( N^{(2p - 1)} \) is 4.

2. **Identifying the Unit Digits**:
Let's analyze the unit digits of numbers from 0 to 9 when raised to even and odd powers:
- **Unit digit 0**: \( 0^{even} = 0 \), \( 0^{odd} = 0 \)
- **Unit digit 1**: \( 1^{even} = 1 \), \( 1^{odd} = 1 \)
- **Unit digit 2**: \( 2^{even} = 6 \), \( 2^{odd} = 2 \)
- **Unit digit 3**: \( 3^{even} = 9 \), \( 3^{odd} = 3 \)
- **Unit digit 4**: \( 4^{even} = 6 \), \( 4^{odd} = 4 \)
- **Unit digit 5**: \( 5^{even} = 5 \), \( 5^{odd} = 5 \)
- **Unit digit 6**: \( 6^{even} = 6 \), \( 6^{odd} = 6 \)
- **Unit digit 7**: \( 7^{even} = 9 \), \( 7^{odd} = 7 \)
- **Unit digit 8**: \( 8^{even} = 4 \), \( 8^{odd} = 8 \)
- **Unit digit 9**: \( 9^{even} = 1 \), \( 9^{odd} = 9 \)

3. **Finding Suitable Unit Digits**:
From our analysis:
- For \( N^{2p} \) to end with 6, \( N \) can have unit digits of 2, 4, or 6.
- For \( N^{(2p - 1)} \) to end with 4, \( N \) must have a unit digit of 4.

4. **Identifying the Greatest Two-Digit Number**:
The greatest two-digit number with a unit digit of 4 is 94.
We will check if \( N = 94 \) satisfies the conditions.

5. **Calculating Unit Digits for \( N = 94 \)**:
- The unit digit of 94 is 4.
- For \( N^{2p} \) (even power), the unit digit is 6 (since \( 4^{even} = 6 \)).
- For \( N^{(2p - 1)} \) (odd power), the unit digit is 4 (since \( 4^{odd} = 4 \)).

6. **Conclusion**:
Since both conditions are satisfied, the greatest two-digit number \( N \) that meets the criteria is **94**.

### Final Answer:
The greatest two-digit number that follows the aforementioned condition is **94**.