Find the unit digit in `8^(96) - 4^(16)`.

Options

A. 0
B. 6
C. 4
D. 2

Answer

To find the unit digit of \( 8^{96} - 4^{16} \), we can follow these steps:

### Step 1: Find the unit digit of \( 8^{96} \)

The unit digits of powers of 8 follow a pattern:
- \( 8^1 = 8 \) (unit digit = 8)
- \( 8^2 = 64 \) (unit digit = 4)
- \( 8^3 = 512 \) (unit digit = 2)
- \( 8^4 = 4096 \) (unit digit = 6)
- \( 8^5 = 32768 \) (unit digit = 8) and the pattern repeats every 4 terms: 8, 4, 2, 6.

To find the unit digit of \( 8^{96} \), we can divide the exponent by 4:
\[ 96 \div 4 = 24 \quad \text{(remainder 0)} \]
Since the remainder is 0, the unit digit corresponds to \( 8^4 \), which is 6.

### Step 2: Find the unit digit of \( 4^{16} \)

The unit digits of powers of 4 also follow a pattern:
- \( 4^1 = 4 \) (unit digit = 4)
- \( 4^2 = 16 \) (unit digit = 6)
- \( 4^3 = 64 \) (unit digit = 4)
- \( 4^4 = 256 \) (unit digit = 6) and the pattern repeats every 2 terms: 4, 6.

To find the unit digit of \( 4^{16} \), we can divide the exponent by 2:
\[ 16 \div 2 = 8 \quad \text{(remainder 0)} \]
Since the remainder is 0, the unit digit corresponds to \( 4^2 \), which is 6.

### Step 3: Calculate the unit digit of \( 8^{96} - 4^{16} \)

Now, we have:
- Unit digit of \( 8^{96} = 6 \)
- Unit digit of \( 4^{16} = 6 \)

Now, we subtract the unit digits:
\[ 6 - 6 = 0 \]

### Final Answer

The unit digit of \( 8^{96} - 4^{16} \) is **0**.