If n is a natural number, then which number always ends at 6 from the following ?

Options

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

Explanation

To determine which number always ends with the digit 6 when raised to the power of a natural number \( n \), we will analyze each option provided:

### Step-by-Step Solution:

1. **Identify the Options**: The options given are:
- (A) \( 4^n \)
- (B) \( 2^n \)
- (C) \( 6^n \)
- (D) \( 8^n \)

2. **Check Option (A): \( 4^n \)**:
- Calculate the first few powers of 4:
- \( 4^1 = 4 \) (ends with 4)
- \( 4^2 = 16 \) (ends with 6)
- \( 4^3 = 64 \) (ends with 4)
- \( 4^4 = 256 \) (ends with 6)
- The units digit alternates between 4 and 6, so it does not always end with 6.

**Conclusion**: Option (A) is eliminated.

3. **Check Option (B): \( 2^n \)**:
- Calculate the first few powers of 2:
- \( 2^1 = 2 \) (ends with 2)
- \( 2^2 = 4 \) (ends with 4)
- \( 2^3 = 8 \) (ends with 8)
- \( 2^4 = 16 \) (ends with 6)
- \( 2^5 = 32 \) (ends with 2)
- The units digit cycles through 2, 4, 8, 6, so it does not always end with 6.

**Conclusion**: Option (B) is eliminated.

4. **Check Option (C): \( 6^n \)**:
- Calculate the first few powers of 6:
- \( 6^1 = 6 \) (ends with 6)
- \( 6^2 = 36 \) (ends with 6)
- \( 6^3 = 216 \) (ends with 6)
- \( 6^4 = 1296 \) (ends with 6)
- The units digit is always 6 regardless of the value of \( n \).

**Conclusion**: Option (C) is valid.

5. **Check Option (D): \( 8^n \)**:
- Calculate the first few powers of 8:
- \( 8^1 = 8 \) (ends with 8)
- \( 8^2 = 64 \) (ends with 4)
- \( 8^3 = 512 \) (ends with 2)
- \( 8^4 = 4096 \) (ends with 6)
- The units digit cycles through 8, 4, 2, 6, so it does not always end with 6.

**Conclusion**: Option (D) is eliminated.

### Final Answer:
The only option that always ends with the digit 6 is **(C) \( 6^n \)**.