Question

For any natural number \(n\), \(6^{n}\) ends with the digit :

• A. 0
• B. 6
• C. 3
• D. 2

Hint:

The digits 0, 1, 5, and 6 always result in the same unit digit (0, 1, 5, and 6 respectively) when raised to any positive integer power.
For example, \(5^{n}\) always ends in 5, and \(6^{n}\) always ends in 6.

Answer 1

The Correct Option is B

To determine the last digit of \(6^n\) for any natural number \(n\), we need to examine the pattern of the last digits of the powers of 6.

Let's compute the first few powers of 6 to observe the pattern:

1. \(6^1 = 6\) – The last digit is 6.
2. \(6^2 = 36\) – The last digit is 6.
3. \(6^3 = 216\) – The last digit is 6.
4. \(6^4 = 1296\) – The last digit is 6.

From the calculations above, we can see a clear pattern: the last digit of \(6^n\) is always 6 for any positive integer \(n\).

Here's why this pattern occurs:

The powers of any number ending in 6 will continue to end in 6, because multiplying a number ending in 6 by 6 always results in a number whose last digit is 6.

The last digit of any multiplication of numbers where the last digit of one of the multiplicands is 6 will inherit the 6. Therefore, the powers of 6 will always result in a last digit of 6.

Given the options:

• 0
• 6 (Correct answer)
• 3
• 2

Hence, the correct answer is 6, as demonstrated by the consistent pattern observed in the last digits for powers of 6.