Find the unit digit of `(432)^(412) xx (499)^(431)`

Options

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

Answer

To find the unit digit of \( (432)^{412} \times (499)^{431} \), we will break down the problem into manageable steps.

### Step 1: Find the unit digit of \( 432^{412} \)

1. **Identify the unit digit of 432**: The unit digit of 432 is 2.
2. **Determine the cyclicity of the unit digit of 2**: The powers of 2 have a cyclicity of 4:
- \( 2^1 = 2 \) (unit digit 2)
- \( 2^2 = 4 \) (unit digit 4)
- \( 2^3 = 8 \) (unit digit 8)
- \( 2^4 = 16 \) (unit digit 6)
- \( 2^5 = 32 \) (unit digit 2) and so on.
3. **Find \( 412 \mod 4 \)**:
- \( 412 \div 4 = 103 \) remainder \( 0 \).
- Thus, \( 412 \mod 4 = 0 \).
4. **Determine the unit digit for \( 2^{412} \)**: Since \( 412 \mod 4 = 0 \), the unit digit corresponds to \( 2^4 \), which is 6.

### Step 2: Find the unit digit of \( 499^{431} \)

1. **Identify the unit digit of 499**: The unit digit of 499 is 9.
2. **Determine the cyclicity of the unit digit of 9**: The powers of 9 have a cyclicity of 2:
- \( 9^1 = 9 \) (unit digit 9)
- \( 9^2 = 81 \) (unit digit 1)
- \( 9^3 = 729 \) (unit digit 9) and so on.
3. **Find \( 431 \mod 2 \)**:
- \( 431 \div 2 = 215 \) remainder \( 1 \).
- Thus, \( 431 \mod 2 = 1 \).
4. **Determine the unit digit for \( 9^{431} \)**: Since \( 431 \mod 2 = 1 \), the unit digit corresponds to \( 9^1 \), which is 9.

### Step 3: Combine the unit digits

Now we have:
- The unit digit of \( 432^{412} \) is 6.
- The unit digit of \( 499^{431} \) is 9.

To find the unit digit of the product \( (432^{412} \times 499^{431}) \):
- Calculate \( 6 \times 9 = 54 \).
- The unit digit of 54 is 4.

### Final Answer

Thus, the unit digit of \( (432)^{412} \times (499)^{431} \) is **4**.