Find the unit place digit in `(194)^(102)+(294)^(103)`

Options

A. 0
B. 6
C. 8
D. 2

Answer

To find the unit place digit in the expression \( (194)^{102} + (294)^{103} \), we can follow these steps:

### Step 1: Identify the unit digits of the bases
The unit digit of \( 194 \) is \( 4 \) and the unit digit of \( 294 \) is also \( 4 \).

### Step 2: Analyze the powers of the unit digit
We need to find the unit digit of \( 4^{102} \) and \( 4^{103} \).

### Step 3: Determine the pattern of unit digits for powers of 4
The unit digits of powers of \( 4 \) follow a specific pattern:
- \( 4^1 = 4 \) (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:
- If the power is odd, the unit digit is \( 4 \).
- If the power is even, the unit digit is \( 6 \).

### Step 4: Calculate the unit digits for \( 4^{102} \) and \( 4^{103} \)
- Since \( 102 \) is even, the unit digit of \( 4^{102} \) is \( 6 \).
- Since \( 103 \) is odd, the unit digit of \( 4^{103} \) is \( 4 \).

### Step 5: Add the unit digits
Now we add the unit digits we found:
\[ 6 + 4 = 10 \]

### Step 6: Find the unit digit of the sum
The unit digit of \( 10 \) is \( 0 \).

### Final Answer
Thus, the unit place digit in \( (194)^{102} + (294)^{103} \) is \( 0 \).

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