Unit digit in `(264)^(102)+(264)^(103)` is :

Options

A. 0
B. 4
C. 6
D. 8

Answer

To find the unit digit of the expression \( (264)^{102} + (264)^{103} \), we can follow these steps:

### Step 1: Identify the unit digit of the base
The base of both terms is 264. The unit digit of 264 is 4.

### Step 2: Determine the cyclicity of the unit digit of powers of 4
Next, we need to find the pattern in the unit digits of powers of 4:
- \( 4^1 = 4 \) (unit digit is 4)
- \( 4^2 = 16 \) (unit digit is 6)
- \( 4^3 = 64 \) (unit digit is 4)
- \( 4^4 = 256 \) (unit digit is 6)

From this, we can see that the unit digits of powers of 4 alternate:
- If the power is odd, the unit digit is 4.
- If the power is even, the unit digit is 6.

### Step 3: Analyze the powers in the expression
Now, we analyze the powers in our expression:
- \( 264^{102} \): Since 102 is even, the unit digit will be 6.
- \( 264^{103} \): Since 103 is odd, the unit digit will be 4.

### Step 4: Add the unit digits
Now we add the unit digits obtained from the previous step:
- Unit digit of \( 264^{102} \) is 6.
- Unit digit of \( 264^{103} \) is 4.

Adding these gives:
\[ 6 + 4 = 10 \]

### Step 5: Find the unit digit of the sum
The unit digit of 10 is 0.

### Conclusion
Thus, the unit digit in \( (264)^{102} + (264)^{103} \) is **0**.

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