Find out the unit digit in -
`(23)^(21) xx (24)^(22) xx (26)^(23) xx (27)^(24) xx (25)^(25)`

Options

A. 2
B. 4
C. 0
D. 5

Answer

To find the unit digit of the expression \( (23)^{21} \times (24)^{22} \times (26)^{23} \times (27)^{24} \times (25)^{25} \), we will focus on the unit digits of each base and their respective powers.

### Step 1: Identify the unit digits of the bases
- The unit digit of \( 23 \) is \( 3 \).
- The unit digit of \( 24 \) is \( 4 \).
- The unit digit of \( 26 \) is \( 6 \).
- The unit digit of \( 27 \) is \( 7 \).
- The unit digit of \( 25 \) is \( 5 \).

### Step 2: Calculate the unit digit of each term
1. **For \( (23)^{21} \) or \( (3)^{21} \)**:
- The unit digits of powers of \( 3 \) cycle every 4:
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 7 \)
- \( 3^4 = 1 \)
- Since \( 21 \mod 4 = 1 \), the unit digit is \( 3 \).

2. **For \( (24)^{22} \) or \( (4)^{22} \)**:
- The unit digits of powers of \( 4 \) cycle every 2:
- \( 4^1 = 4 \)
- \( 4^2 = 6 \)
- Since \( 22 \mod 2 = 0 \), the unit digit is \( 6 \).

3. **For \( (26)^{23} \) or \( (6)^{23} \)**:
- The unit digit of \( 6 \) raised to any power is always \( 6 \).

4. **For \( (27)^{24} \) or \( (7)^{24} \)**:
- The unit digits of powers of \( 7 \) cycle every 4:
- \( 7^1 = 7 \)
- \( 7^2 = 9 \)
- \( 7^3 = 3 \)
- \( 7^4 = 1 \)
- Since \( 24 \mod 4 = 0 \), the unit digit is \( 1 \).

5. **For \( (25)^{25} \) or \( (5)^{25} \)**:
- The unit digit of \( 5 \) raised to any power is always \( 5 \).

### Step 3: Combine the unit digits
Now we have the unit digits:
- From \( (23)^{21} \): \( 3 \)
- From \( (24)^{22} \): \( 6 \)
- From \( (26)^{23} \): \( 6 \)
- From \( (27)^{24} \): \( 1 \)
- From \( (25)^{25} \): \( 5 \)

### Step 4: Calculate the final unit digit
Now we multiply the unit digits:
\[ 3 \times 6 = 18 \quad \text{(unit digit is 8)} \]
\[ 8 \times 6 = 48 \quad \text{(unit digit is 8)} \]
\[ 8 \times 1 = 8 \quad \text{(unit digit is 8)} \]
\[ 8 \times 5 = 40 \quad \text{(unit digit is 0)} \]

### Final Answer
The unit digit of the entire expression \( (23)^{21} \times (24)^{22} \times (26)^{23} \times (27)^{24} \times (25)^{25} \) is \( \boxed{0} \).