`259^(148)-123^(43)` Find Unit digit

Answer

To find the unit digit of the expression \( 259^{148} - 123^{43} \), we can follow these steps:

### Step 1: Identify the unit digits of the bases
The unit digit of \( 259 \) is \( 9 \) and the unit digit of \( 123 \) is \( 3 \).

### Step 2: Determine the unit digit of \( 259^{148} \)
We need to find the unit digit of \( 9^{148} \). The unit digits of powers of \( 9 \) follow a pattern:
- \( 9^1 = 9 \) (unit digit is \( 9 \))
- \( 9^2 = 81 \) (unit digit is \( 1 \))
- \( 9^3 = 729 \) (unit digit is \( 9 \))
- \( 9^4 = 6561 \) (unit digit is \( 1 \))

The pattern repeats every 2 terms: \( 9, 1 \).

To find the unit digit of \( 9^{148} \), we can determine \( 148 \mod 2 \):
- \( 148 \div 2 = 74 \) with a remainder of \( 0 \).

Since the remainder is \( 0 \), the unit digit of \( 9^{148} \) corresponds to \( 9^2 \), which is \( 1 \).

### Step 3: Determine the unit digit of \( 123^{43} \)
Next, we find the unit digit of \( 3^{43} \). The unit digits of powers of \( 3 \) follow a pattern:
- \( 3^1 = 3 \) (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 pattern repeats every 4 terms: \( 3, 9, 7, 1 \).

To find the unit digit of \( 3^{43} \), we can determine \( 43 \mod 4 \):
- \( 43 \div 4 = 10 \) with a remainder of \( 3 \).

Since the remainder is \( 3 \), the unit digit of \( 3^{43} \) corresponds to \( 3^3 \), which is \( 7 \).

### Step 4: Calculate the unit digit of \( 259^{148} - 123^{43} \)
Now we have:
- Unit digit of \( 259^{148} \) is \( 1 \).
- Unit digit of \( 123^{43} \) is \( 7 \).

We need to calculate the unit digit of \( 1 - 7 \):
- \( 1 - 7 = -6 \).

Since we are looking for the unit digit, we can convert \( -6 \) to a positive unit digit by adding \( 10 \):
- \( -6 + 10 = 4 \).

### Final Answer
The unit digit of \( 259^{148} - 123^{43} \) is \( 4 \).