What is the ones ( or unit ) digit in `23^(23) - 17^(17)` ?

Options

A. 0
B. 2
C. 3
D. 7

Answer

To find the unit digit of \( 23^{23} - 17^{17} \), we can follow these steps:

### Step 1: Find the unit digit of \( 23^{23} \)

1. **Identify the unit digit of the base**: The unit digit of \( 23 \) is \( 3 \).
2. **Determine the cycle of unit digits for powers of \( 3 \)**:
- \( 3^1 = 3 \) (unit digit is \( 3 \))
- \( 3^2 = 9 \) (unit digit is \( 9 \))
- \( 3^3 = 27 \) (unit digit is \( 7 \))
- \( 3^4 = 81 \) (unit digit is \( 1 \))
- The cycle repeats every 4: \( 3, 9, 7, 1 \).

3. **Find the exponent modulo 4**:
- Calculate \( 23 \mod 4 \):
\[ 23 \div 4 = 5 \quad \text{(remainder } 3\text{)} \]
- So, \( 23 \mod 4 = 3 \).

4. **Use the cycle to find the unit digit**:
- The third position in the cycle \( (3, 9, 7, 1) \) corresponds to \( 7 \).
- Therefore, the unit digit of \( 23^{23} \) is \( 7 \).

### Step 2: Find the unit digit of \( 17^{17} \)

1. **Identify the unit digit of the base**: The unit digit of \( 17 \) is \( 7 \).
2. **Determine the cycle of unit digits for powers of \( 7 \)**:
- \( 7^1 = 7 \) (unit digit is \( 7 \))
- \( 7^2 = 49 \) (unit digit is \( 9 \))
- \( 7^3 = 343 \) (unit digit is \( 3 \))
- \( 7^4 = 2401 \) (unit digit is \( 1 \))
- The cycle repeats every 4: \( 7, 9, 3, 1 \).

3. **Find the exponent modulo 4**:
- Calculate \( 17 \mod 4 \):
\[ 17 \div 4 = 4 \quad \text{(remainder } 1\text{)} \]
- So, \( 17 \mod 4 = 1 \).

4. **Use the cycle to find the unit digit**:
- The first position in the cycle \( (7, 9, 3, 1) \) corresponds to \( 7 \).
- Therefore, the unit digit of \( 17^{17} \) is \( 7 \).

### Step 3: Calculate the unit digit of \( 23^{23} - 17^{17} \)

1. **Subtract the unit digits**:
- Unit digit of \( 23^{23} \) is \( 7 \).
- Unit digit of \( 17^{17} \) is \( 7 \).
- Therefore, we calculate:
\[ 7 - 7 = 0 \]

### Conclusion

The unit digit of \( 23^{23} - 17^{17} \) is \( 0 \).

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