Find the unit digit in `(6^(36))^(84)`.

Options

A. 6
B. 4
C. 7
D. 1

Answer

To find the unit digit of \((6^{36})^{84}\), we can simplify the expression and use properties of powers and modular arithmetic.

### Step-by-step Solution:

1. **Simplify the Expression**:
\[ (6^{36})^{84} = 6^{36 \times 84} = 6^{3024} \]

2. **Determine the Unit Digit**:
To find the unit digit of \(6^{3024}\), we observe the pattern of unit digits in powers of 6:
- \(6^1 = 6\) (unit digit is 6)
- \(6^2 = 36\) (unit digit is 6)
- \(6^3 = 216\) (unit digit is 6)
- \(6^4 = 1296\) (unit digit is 6)

From this, we can see that the unit digit of \(6^n\) for any positive integer \(n\) is always 6.

3. **Conclusion**:
Since the unit digit of \(6^{3024}\) is the same as the unit digit of \(6^1\), we conclude that the unit digit of \((6^{36})^{84}\) is:
\[ \text{Unit digit} = 6 \]

### Final Answer:
The unit digit in \((6^{36})^{84}\) is **6**.