Find the unit digit of expression
`(823)^(933!) × (777)^(223!) × (838)^(123!) × (525)^(111!)`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the unit digit of the expression \( (823)^{933!} \times (777)^{223!} \times (838)^{123!} \times (525)^{111!} \), we will analyze the unit digits of each base number and how they behave under exponentiation.

### Step 1: Identify the unit digits of the base numbers
- The unit digit of \( 823 \) is \( 3 \).
- The unit digit of \( 777 \) is \( 7 \).
- The unit digit of \( 838 \) is \( 8 \).
- The unit digit of \( 525 \) is \( 5 \).

### Step 2: Analyze the unit digit of each term
1. **For \( (823)^{933!} \)**:
- The unit digit of powers of \( 3 \) cycles every 4: \( 3, 9, 7, 1 \).
- To find the relevant power, calculate \( 933! \mod 4 \).
- Since \( 933! \) is a factorial of a number greater than \( 4 \), it will be \( 0 \mod 4 \).
- Therefore, the unit digit of \( (823)^{933!} \) is \( 1 \) (from the cycle).

2. **For \( (777)^{223!} \)**:
- The unit digit of powers of \( 7 \) cycles every 4: \( 7, 9, 3, 1 \).
- Calculate \( 223! \mod 4 \).
- Since \( 223! \) is also a factorial of a number greater than \( 4 \), it will be \( 0 \mod 4 \).
- Therefore, the unit digit of \( (777)^{223!} \) is \( 1 \).

3. **For \( (838)^{123!} \)**:
- The unit digit of powers of \( 8 \) cycles every 4: \( 8, 4, 2, 6 \).
- Calculate \( 123! \mod 4 \).
- Since \( 123! \) is also a factorial of a number greater than \( 4 \), it will be \( 0 \mod 4 \).
- Therefore, the unit digit of \( (838)^{123!} \) is \( 6 \).

4. **For \( (525)^{111!} \)**:
- The unit digit of \( 5 \) raised to any positive integer power is always \( 5 \).
- Therefore, the unit digit of \( (525)^{111!} \) is \( 5 \).

### Step 3: Combine the unit digits
Now we combine the unit digits of all the terms:
- From \( (823)^{933!} \) we have \( 1 \).
- From \( (777)^{223!} \) we have \( 1 \).
- From \( (838)^{123!} \) we have \( 6 \).
- From \( (525)^{111!} \) we have \( 5 \).

So we need to calculate:
\[ 1 \times 1 \times 6 \times 5 \]

### Step 4: Calculate the final unit digit
Calculating this step-by-step:
1. \( 1 \times 1 = 1 \)
2. \( 1 \times 6 = 6 \)
3. \( 6 \times 5 = 30 \)

The unit digit of \( 30 \) is \( 0 \).

### Conclusion
The unit digit of the expression \( (823)^{933!} \times (777)^{223!} \times (838)^{123!} \times (525)^{111!} \) is \( 0 \).