The unit digit of `(12345k)^(72)` is 6. The value of k is :

Options

A. 6
B. 8
C. 2
D. all of these

Answer

To find the value of \( k \) such that the unit digit of \( (12345k)^{72} \) is 6, we will focus on the unit digit of \( k \) raised to the power of 72.

### Step-by-Step Solution:

1. **Identify the Unit Digit**:
The unit digit of \( 12345k \) is determined solely by the unit digit of \( k \) because the unit digit of \( 12345 \) is 5. Therefore, we need to find the unit digit of \( k^{72} \).

2. **Determine Possible Values of \( k \)**:
The unit digit of \( k^{72} \) must equal 6. We will check the unit digits of \( k \) from 0 to 9 to see which ones can yield a unit digit of 6 when raised to the power of 72.

3. **Calculate Unit Digits for Each \( k \)**:
- \( k = 0 \): \( 0^{72} = 0 \) (unit digit is 0)
- \( k = 1 \): \( 1^{72} = 1 \) (unit digit is 1)
- \( k = 2 \): The unit digits cycle through 2, 4, 8, 6. Since \( 72 \mod 4 = 0 \), the unit digit is 6.
- \( k = 3 \): The unit digits cycle through 3, 9, 7, 1. Since \( 72 \mod 4 = 0 \), the unit digit is 1.
- \( k = 4 \): The unit digits cycle through 4, 6. Since \( 72 \mod 2 = 0 \), the unit digit is 6.
- \( k = 5 \): \( 5^{72} = 5 \) (unit digit is 5)
- \( k = 6 \): \( 6^{72} = 6 \) (unit digit is 6)
- \( k = 7 \): The unit digits cycle through 7, 9, 3, 1. Since \( 72 \mod 4 = 0 \), the unit digit is 1.
- \( k = 8 \): The unit digits cycle through 8, 4, 2, 6. Since \( 72 \mod 4 = 0 \), the unit digit is 6.
- \( k = 9 \): The unit digits cycle through 9, 1. Since \( 72 \mod 2 = 0 \), the unit digit is 1.

4. **List Valid Values of \( k \)**:
From the calculations, the values of \( k \) that yield a unit digit of 6 when raised to the power of 72 are:
- \( k = 2 \)
- \( k = 4 \)
- \( k = 6 \)
- \( k = 8 \)

5. **Conclusion**:
Therefore, the possible values of \( k \) that satisfy the condition are \( 2, 4, 6, \) and \( 8 \).

### Final Answer:
The values of \( k \) are \( 2, 4, 6, \) and \( 8 \).

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