Find the unit digit of expression
`(259)^123 – (525)^111 – (236)^122 – (414)^115 + (323)^81`

Options

A. 3
B. 4
C. 5
D. 7

Answer

To find the unit digit of the expression \( (259)^{123} - (525)^{111} - (236)^{122} - (414)^{115} + (323)^{81} \), we will analyze the unit digits of each term separately.

### Step 1: Find the unit digit of \( (259)^{123} \)
- The unit digit of \( 259 \) is \( 9 \).
- The powers of \( 9 \) cycle through \( 9, 1 \) (i.e., \( 9^1 = 9 \), \( 9^2 = 81 \) (unit digit \( 1 \)), \( 9^3 = 729 \) (unit digit \( 9 \)), \( 9^4 = 6561 \) (unit digit \( 1 \)), ...).
- The cycle length is \( 2 \).
- To find \( 9^{123} \), we calculate \( 123 \mod 2 = 1 \).
- Thus, the unit digit of \( (259)^{123} \) is \( 9 \).

### Step 2: Find the unit digit of \( (525)^{111} \)
- The unit digit of \( 525 \) is \( 5 \).
- The unit digit of any power of \( 5 \) is always \( 5 \).
- Thus, the unit digit of \( (525)^{111} \) is \( 5 \).

### Step 3: Find the unit digit of \( (236)^{122} \)
- The unit digit of \( 236 \) is \( 6 \).
- The unit digit of any power of \( 6 \) is always \( 6 \).
- Thus, the unit digit of \( (236)^{122} \) is \( 6 \).

### Step 4: Find the unit digit of \( (414)^{115} \)
- The unit digit of \( 414 \) is \( 4 \).
- The powers of \( 4 \) cycle through \( 4, 6 \) (i.e., \( 4^1 = 4 \), \( 4^2 = 16 \) (unit digit \( 6 \)), \( 4^3 = 64 \) (unit digit \( 4 \)), \( 4^4 = 256 \) (unit digit \( 6 \)), ...).
- The cycle length is \( 2 \).
- To find \( 4^{115} \), we calculate \( 115 \mod 2 = 1 \).
- Thus, the unit digit of \( (414)^{115} \) is \( 4 \).

### Step 5: Find the unit digit of \( (323)^{81} \)
- The unit digit of \( 323 \) is \( 3 \).
- The powers of \( 3 \) cycle through \( 3, 9, 7, 1 \) (i.e., \( 3^1 = 3 \), \( 3^2 = 9 \), \( 3^3 = 27 \) (unit digit \( 7 \)), \( 3^4 = 81 \) (unit digit \( 1 \)), ...).
- The cycle length is \( 4 \).
- To find \( 3^{81} \), we calculate \( 81 \mod 4 = 1 \).
- Thus, the unit digit of \( (323)^{81} \) is \( 3 \).

### Step 6: Combine the unit digits
Now we can substitute the unit digits back into the expression:
- \( 9 - 5 - 6 - 4 + 3 \)

Calculating this step-by-step:
1. \( 9 - 5 = 4 \)
2. \( 4 - 6 = -2 \)
3. \( -2 - 4 = -6 \)
4. \( -6 + 3 = -3 \)

### Step 7: Find the final unit digit
- The result is \( -3 \).
- To convert this to a positive unit digit, we can add \( 10 \) (since we are interested in the unit digit):
- \( -3 + 10 = 7 \)

Thus, the unit digit of the expression \( (259)^{123} - (525)^{111} - (236)^{122} - (414)^{115} + (323)^{81} \) is **7**.