Find the digit at the unit's place of
`(377)^(59)xx(793)^(87)xx(578)^(129)xx(99)^(99)`

Options

A. 1
B. 2
C. 7
D. 9

Answer

To find the digit at the unit's place of the expression \( (377)^{59} \times (793)^{87} \times (578)^{129} \times (99)^{99} \), we can follow these steps:

### Step 1: Identify the unit digits of each base
- The unit digit of \( 377 \) is \( 7 \).
- The unit digit of \( 793 \) is \( 3 \).
- The unit digit of \( 578 \) is \( 8 \).
- The unit digit of \( 99 \) is \( 9 \).

### Step 2: Rewrite the expression using only the unit digits
We can rewrite the expression focusing on the unit digits:
\[ (7^{59}) \times (3^{87}) \times (8^{129}) \times (9^{99}) \]

### Step 3: Determine the cyclicity of the unit digits
- The unit digit of \( 7 \) has a cycle of \( 4 \): \( 7, 9, 3, 1 \).
- The unit digit of \( 3 \) also has a cycle of \( 4 \): \( 3, 9, 7, 1 \).
- The unit digit of \( 8 \) has a cycle of \( 4 \): \( 8, 4, 2, 6 \).
- The unit digit of \( 9 \) has a cycle of \( 2 \): \( 9, 1 \) (odd powers yield \( 9 \), even powers yield \( 1 \)).

### Step 4: Calculate the effective powers modulo their cycles
- For \( 7^{59} \):
\[ 59 \mod 4 = 3 \quad \text{(since } 59 = 4 \times 14 + 3\text{)} \]
Thus, the unit digit is \( 7^3 = 343 \) → unit digit is \( 3 \).

- For \( 3^{87} \):
\[ 87 \mod 4 = 3 \quad \text{(since } 87 = 4 \times 21 + 3\text{)} \]
Thus, the unit digit is \( 3^3 = 27 \) → unit digit is \( 7 \).

- For \( 8^{129} \):
\[ 129 \mod 4 = 1 \quad \text{(since } 129 = 4 \times 32 + 1\text{)} \]
Thus, the unit digit is \( 8^1 = 8 \).

- For \( 9^{99} \):
Since \( 99 \) is odd, the unit digit is \( 9 \).

### Step 5: Multiply the unit digits together
Now we multiply the unit digits we found:
\[ 3 \times 7 \times 8 \times 9 \]

Calculating step by step:
1. \( 3 \times 7 = 21 \) → unit digit is \( 1 \).
2. \( 1 \times 8 = 8 \).
3. \( 8 \times 9 = 72 \) → unit digit is \( 2 \).

### Conclusion
The unit digit of the entire expression \( (377)^{59} \times (793)^{87} \times (578)^{129} \times (99)^{99} \) is \( 2 \).