What is the unit digit of `(217)^(413)xx(819)^(547)xx(414)^(624)xx(342)^(812)` ?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the unit digit of the expression \( (217)^{413} \times (819)^{547} \times (414)^{624} \times (342)^{812} \), we can simplify the problem by focusing on the unit digits of each base number and their respective powers.

### Step-by-Step Solution:

1. **Identify the Unit Digits:**
- The unit digit of \( 217 \) is \( 7 \).
- The unit digit of \( 819 \) is \( 9 \).
- The unit digit of \( 414 \) is \( 4 \).
- The unit digit of \( 342 \) is \( 2 \).

Thus, we need to find the unit digit of \( (7)^{413} \times (9)^{547} \times (4)^{624} \times (2)^{812} \).

2. **Find the Cyclicity of Each Unit Digit:**
- **For \( 7 \):** The unit digits of powers of \( 7 \) cycle every \( 4 \):
- \( 7^1 = 7 \)
- \( 7^2 = 49 \) (unit digit \( 9 \))
- \( 7^3 = 343 \) (unit digit \( 3 \))
- \( 7^4 = 2401 \) (unit digit \( 1 \))
- The cycle is \( 7, 9, 3, 1 \).

- **For \( 9 \):** The unit digits of powers of \( 9 \) cycle every \( 2 \):
- \( 9^1 = 9 \)
- \( 9^2 = 81 \) (unit digit \( 1 \))
- The cycle is \( 9, 1 \).

- **For \( 4 \):** The unit digits of powers of \( 4 \) cycle every \( 2 \):
- \( 4^1 = 4 \)
- \( 4^2 = 16 \) (unit digit \( 6 \))
- The cycle is \( 4, 6 \).

- **For \( 2 \):** The unit digits of powers of \( 2 \) cycle every \( 4 \):
- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)
- \( 2^4 = 16 \) (unit digit \( 6 \))
- The cycle is \( 2, 4, 8, 6 \).

3. **Calculate the Remainders:**
- For \( 7^{413} \):
- \( 413 \mod 4 = 1 \) (the remainder is \( 1 \)), so the unit digit is \( 7 \).
- For \( 9^{547} \):
- \( 547 \mod 2 = 1 \) (the remainder is \( 1 \)), so the unit digit is \( 9 \).
- For \( 4^{624} \):
- \( 624 \mod 2 = 0 \) (the remainder is \( 0 \)), so the unit digit is \( 6 \).
- For \( 2^{812} \):
- \( 812 \mod 4 = 0 \) (the remainder is \( 0 \)), so the unit digit is \( 6 \).

4. **Combine the Unit Digits:**
- Now we multiply the unit digits obtained:
- \( 7 \times 9 \times 6 \times 6 \).

5. **Calculate Step by Step:**
- First, calculate \( 7 \times 9 = 63 \) (unit digit \( 3 \)).
- Next, calculate \( 3 \times 6 = 18 \) (unit digit \( 8 \)).
- Finally, calculate \( 8 \times 6 = 48 \) (unit digit \( 8 \)).

Thus, the unit digit of \( (217)^{413} \times (819)^{547} \times (414)^{624} \times (342)^{812} \) is **8**.