Find out the unit digit in -
`(235)^(215) + (314)^(326) + (6736)^(213) + (3167)^(112)`

Options

A. 2
B. 0
C. 8
D. 5

Answer

To find the unit digit of the expression \( (235)^{215} + (314)^{326} + (6736)^{213} + (3167)^{112} \), we can simplify our calculations by focusing only on the unit digits of each term in the expression.

### Step-by-step Solution:

1. **Identify the unit digits of the bases:**
- The unit digit of \( 235 \) is \( 5 \).
- The unit digit of \( 314 \) is \( 4 \).
- The unit digit of \( 6736 \) is \( 6 \).
- The unit digit of \( 3167 \) is \( 7 \).

2. **Calculate the unit digit of \( (235)^{215} \):**
- Any power of \( 5 \) ends with \( 5 \).
- Therefore, the unit digit of \( (235)^{215} \) is \( 5 \).

3. **Calculate the unit digit of \( (314)^{326} \):**
- The unit digit of \( 4 \) follows a pattern:
- \( 4^1 = 4 \)
- \( 4^2 = 16 \) (unit digit is \( 6 \))
- \( 4^3 = 64 \) (unit digit is \( 4 \))
- \( 4^4 = 256 \) (unit digit is \( 6 \))
- The pattern repeats every two terms: \( 4, 6 \).
- Since \( 326 \) is even, the unit digit of \( (314)^{326} \) is \( 6 \).

4. **Calculate the unit digit of \( (6736)^{213} \):**
- Any power of \( 6 \) ends with \( 6 \).
- Therefore, the unit digit of \( (6736)^{213} \) is \( 6 \).

5. **Calculate the unit digit of \( (3167)^{112} \):**
- The unit digit of \( 7 \) follows a pattern:
- \( 7^1 = 7 \)
- \( 7^2 = 49 \) (unit digit is \( 9 \))
- \( 7^3 = 343 \) (unit digit is \( 3 \))
- \( 7^4 = 2401 \) (unit digit is \( 1 \))
- The pattern repeats every four terms: \( 7, 9, 3, 1 \).
- To find the unit digit for \( 112 \), we take \( 112 \mod 4 \), which gives \( 0 \).
- Hence, the unit digit of \( (3167)^{112} \) corresponds to \( 7^4 \), which is \( 1 \).

6. **Sum the unit digits:**
- Now we add the unit digits we found:
- \( 5 + 6 + 6 + 1 = 18 \).

7. **Find the unit digit of the sum:**
- The unit digit of \( 18 \) is \( 8 \).

### Final Answer:
The unit digit of \( (235)^{215} + (314)^{326} + (6736)^{213} + (3167)^{112} \) is \( 8 \).