Find the unit place digit in the expression `(159)^(144)+(123)-(110)^(510)+(213)^(25)`

Options

A. 3
B. 4
C. 7
D. 8

Answer

To find the unit place digit in the expression \( (159)^{144} + (123) - (110)^{510} + (213)^{25} \), we will analyze each term separately and then combine the results.

### Step 1: Find the unit digit of \( (159)^{144} \)

1. The unit digit of \( 159 \) is \( 9 \).
2. The unit digits of powers of \( 9 \) follow a pattern:
- \( 9^1 = 9 \) (unit digit is 9)
- \( 9^2 = 81 \) (unit digit is 1)
- \( 9^3 = 729 \) (unit digit is 9)
- \( 9^4 = 6561 \) (unit digit is 1)

The pattern repeats every 2 terms: \( 9, 1 \).

3. Since \( 144 \) is even, the unit digit of \( 9^{144} \) is the same as \( 9^2 \), which is \( 1 \).

### Step 2: Find the unit digit of \( (123) \)

1. The unit digit of \( 123 \) is \( 3 \).
2. The unit digits of powers of \( 3 \) follow a pattern:
- \( 3^1 = 3 \) (unit digit is 3)
- \( 3^2 = 9 \) (unit digit is 9)
- \( 3^3 = 27 \) (unit digit is 7)
- \( 3^4 = 81 \) (unit digit is 1)

The pattern repeats every 4 terms: \( 3, 9, 7, 1 \).

3. To find the unit digit of \( 3^1 \) (since \( 123 \) is not raised to any power), the unit digit is \( 3 \).

### Step 3: Find the unit digit of \( (110)^{510} \)

1. The unit digit of \( 110 \) is \( 0 \).
2. Any power of \( 0 \) is \( 0 \), so the unit digit of \( (110)^{510} \) is \( 0 \).

### Step 4: Find the unit digit of \( (213)^{25} \)

1. The unit digit of \( 213 \) is \( 3 \).
2. We already established the pattern for \( 3 \) in Step 2.
3. To find the unit digit of \( 3^{25} \), we calculate \( 25 \mod 4 \) (since the pattern repeats every 4 terms):
- \( 25 \div 4 = 6 \) remainder \( 1 \).
- Thus, \( 3^{25} \) has the same unit digit as \( 3^1 \), which is \( 3 \).

### Step 5: Combine the results

Now we combine the unit digits from each part of the expression:

- From \( (159)^{144} \): \( 1 \)
- From \( (123) \): \( 3 \)
- From \( (110)^{510} \): \( 0 \)
- From \( (213)^{25} \): \( 3 \)

Putting it all together:
\[ 1 + 3 - 0 + 3 = 7 \]

### Final Answer
The unit place digit in the expression \( (159)^{144} + (123) - (110)^{510} + (213)^{25} \) is \( 7 \).