Find the unit place digit in `(82)^(102)+(183)^(103)`

Options

A. 1
B. 6
C. 8
D. 9

Answer

To find the unit place digit of \( (82)^{102} + (183)^{103} \), we can follow these steps:

### Step 1: Find the unit digit of \( (82)^{102} \)
1. **Identify the unit digit of the base**: The unit digit of \( 82 \) is \( 2 \).
2. **Determine the pattern of unit digits for powers of \( 2 \)**:
- \( 2^1 = 2 \) (unit digit = 2)
- \( 2^2 = 4 \) (unit digit = 4)
- \( 2^3 = 8 \) (unit digit = 8)
- \( 2^4 = 16 \) (unit digit = 6)
- The pattern repeats every 4: \( 2, 4, 8, 6 \).

3. **Find the exponent modulo 4**:
- \( 102 \mod 4 = 2 \) (since \( 102 = 4 \times 25 + 2 \)).
4. **Identify the unit digit from the pattern**: The 2nd position in the pattern \( 2, 4, 8, 6 \) is \( 4 \).

### Step 2: Find the unit digit of \( (183)^{103} \)
1. **Identify the unit digit of the base**: The unit digit of \( 183 \) is \( 3 \).
2. **Determine the pattern of unit digits for powers of \( 3 \)**:
- \( 3^1 = 3 \) (unit digit = 3)
- \( 3^2 = 9 \) (unit digit = 9)
- \( 3^3 = 27 \) (unit digit = 7)
- \( 3^4 = 81 \) (unit digit = 1)
- The pattern repeats every 4: \( 3, 9, 7, 1 \).

3. **Find the exponent modulo 4**:
- \( 103 \mod 4 = 3 \) (since \( 103 = 4 \times 25 + 3 \)).
4. **Identify the unit digit from the pattern**: The 3rd position in the pattern \( 3, 9, 7, 1 \) is \( 7 \).

### Step 3: Add the unit digits
1. **Add the unit digits found**:
- Unit digit of \( (82)^{102} \) is \( 4 \).
- Unit digit of \( (183)^{103} \) is \( 7 \).
- \( 4 + 7 = 11 \).

2. **Find the unit digit of the sum**: The unit digit of \( 11 \) is \( 1 \).

### Final Answer
The unit place digit in \( (82)^{102} + (183)^{103} \) is **1**.

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