Find the unit digit in`(88)^(178)`

Answer

To find the unit digit of \( (88)^{178} \), we can follow these steps:

### Step 1: Identify the unit digit of the base
The unit digit of \( 88 \) is \( 8 \).

### Step 2: Determine the pattern of unit digits for powers of \( 8 \)
We will calculate the unit digits of the first few powers of \( 8 \):
- \( 8^1 = 8 \) (unit digit is \( 8 \))
- \( 8^2 = 64 \) (unit digit is \( 4 \))
- \( 8^3 = 512 \) (unit digit is \( 2 \))
- \( 8^4 = 4096 \) (unit digit is \( 6 \))

Now, we notice a repeating pattern in the unit digits:
- \( 8, 4, 2, 6 \)

### Step 3: Identify the cycle length
The cycle of unit digits repeats every \( 4 \) powers:
- \( 1 \): \( 8 \)
- \( 2 \): \( 4 \)
- \( 3 \): \( 2 \)
- \( 4 \): \( 6 \)

### Step 4: Find the position in the cycle for \( 178 \)
To find out which unit digit corresponds to \( 8^{178} \), we need to find \( 178 \mod 4 \):
\[ 178 \div 4 = 44 \quad \text{(remainder } 2\text{)} \]
So, \( 178 \mod 4 = 2 \).

### Step 5: Determine the unit digit from the cycle
Since \( 178 \mod 4 = 2 \), we look at the second position in our cycle:
- The second position corresponds to the unit digit \( 4 \).

### Step 6: Conclusion
Thus, the unit digit of \( (88)^{178} \) is \( 4 \).

---