Find the unit place digit in the given expression : `(153)^(144)-(115)^(123)-(111)^(510)+(216)^(25)`

Options

A. 1
B. 5
C. 6
D. 3

Answer

To find the unit place digit of the expression \( (153)^{144} - (115)^{123} - (111)^{510} + (216)^{25} \), we will analyze each term separately and determine the unit digit of each.

### Step 1: Find the unit digit of \( (153)^{144} \)
- The unit digit of \( 153 \) is \( 3 \).
- The unit digits of powers of \( 3 \) cycle every \( 4 \) terms:
- \( 3^1 = 3 \) (unit digit \( 3 \))
- \( 3^2 = 9 \) (unit digit \( 9 \))
- \( 3^3 = 27 \) (unit digit \( 7 \))
- \( 3^4 = 81 \) (unit digit \( 1 \))
- Since \( 144 \mod 4 = 0 \), the unit digit of \( 3^{144} \) corresponds to \( 3^4 \), which is \( 1 \).

### Step 2: Find the unit digit of \( (115)^{123} \)
- The unit digit of \( 115 \) is \( 5 \).
- The unit digit of any power of \( 5 \) is always \( 5 \).
- Therefore, the unit digit of \( (115)^{123} \) is \( 5 \).

### Step 3: Find the unit digit of \( (111)^{510} \)
- The unit digit of \( 111 \) is \( 1 \).
- The unit digit of any power of \( 1 \) is always \( 1 \).
- Therefore, the unit digit of \( (111)^{510} \) is \( 1 \).

### Step 4: Find the unit digit of \( (216)^{25} \)
- The unit digit of \( 216 \) is \( 6 \).
- The unit digit of any power of \( 6 \) is always \( 6 \).
- Therefore, the unit digit of \( (216)^{25} \) is \( 6 \).

### Step 5: Combine the results
Now we combine the unit digits we found:
- Unit digit of \( (153)^{144} \) is \( 1 \).
- Unit digit of \( (115)^{123} \) is \( 5 \).
- Unit digit of \( (111)^{510} \) is \( 1 \).
- Unit digit of \( (216)^{25} \) is \( 6 \).

Putting it all together:
\[ 1 - 5 - 1 + 6 \]

Calculating step-by-step:
1. \( 1 - 5 = -4 \)
2. \( -4 - 1 = -5 \)
3. \( -5 + 6 = 1 \)

The final unit digit of the expression \( (153)^{144} - (115)^{123} - (111)^{510} + (216)^{25} \) is \( 1 \).

### Final Answer
The unit place digit is \( 1 \).