Which of these numbers always end with the digit 6.

Options

A. `4^(n)`
B. `2^(n)`
C. `6^(n)`
D. `8^(n)`

Answer

To determine which of the given numbers always ends with the digit 6, we will analyze each option step by step.

### Step 1: Analyze the options
The numbers given are:
1. \( 4^n \)
2. \( 2^n \)
3. \( 6^n \)
4. \( 8^n \)

We need to check the last digit of each of these expressions for different values of \( n \).

### Step 2: Check \( 4^n \)
- For \( n = 1 \):
\[ 4^1 = 4 \quad (\text{last digit is } 4) \]
- For \( n = 2 \):
\[ 4^2 = 16 \quad (\text{last digit is } 6) \]
- For \( n = 3 \):
\[ 4^3 = 64 \quad (\text{last digit is } 4) \]
- For \( n = 4 \):
\[ 4^4 = 256 \quad (\text{last digit is } 6) \]

**Conclusion**: The last digit of \( 4^n \) alternates between 4 and 6. Thus, it does not always end with 6.

### Step 3: Check \( 2^n \)
- For \( n = 1 \):
\[ 2^1 = 2 \quad (\text{last digit is } 2) \]
- For \( n = 2 \):
\[ 2^2 = 4 \quad (\text{last digit is } 4) \]
- For \( n = 3 \):
\[ 2^3 = 8 \quad (\text{last digit is } 8) \]
- For \( n = 4 \):
\[ 2^4 = 16 \quad (\text{last digit is } 6) \]

**Conclusion**: The last digit of \( 2^n \) varies and does not always end with 6.

### Step 4: Check \( 6^n \)
- For \( n = 1 \):
\[ 6^1 = 6 \quad (\text{last digit is } 6) \]
- For \( n = 2 \):
\[ 6^2 = 36 \quad (\text{last digit is } 6) \]
- For \( n = 3 \):
\[ 6^3 = 216 \quad (\text{last digit is } 6) \]
- For \( n = 4 \):
\[ 6^4 = 1296 \quad (\text{last digit is } 6) \]

**Conclusion**: The last digit of \( 6^n \) is always 6 for any positive integer \( n \).

### Step 5: Check \( 8^n \)
- For \( n = 1 \):
\[ 8^1 = 8 \quad (\text{last digit is } 8) \]
- For \( n = 2 \):
\[ 8^2 = 64 \quad (\text{last digit is } 4) \]
- For \( n = 3 \):
\[ 8^3 = 512 \quad (\text{last digit is } 2) \]
- For \( n = 4 \):
\[ 8^4 = 4096 \quad (\text{last digit is } 6) \]

**Conclusion**: The last digit of \( 8^n \) varies and does not always end with 6.

### Final Answer
The only number that always ends with the digit 6 is:
\[ \text{Option C: } 6^n \]