Question

For any natural number $n$, $6^n$ ends with the digit :

• A. 0
• B. 6
• C. 3
• D. 2

Hint:

The numbers ending with digits 0, 1, 5, and 6 have a cyclicity of 1.
This means any positive integral power of these numbers will always end with the same unit digit.
For example, $5^n$ always ends in 5, and $6^n$ always ends in 6.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the units digit of the number $6^n$ for any positive integer (natural number) $n$.
A natural number $n$ belongs to the set $\{1, 2, 3, 4, \dots\}$.
We need to analyze the pattern or the prime factorization of $6^n$ to determine its ending digit.

Step 2: Key Formula or Approach:
For any number to end with the digit 0, its prime factorization must contain both 2 and 5 as prime factors.
For other digits, we can look at the cyclicity of the unit digit under exponentiation.
Alternatively, we can express the number in the form of $10k + r$, where $r$ is the unit digit.

Step 3: Detailed Explanation:

• Let us analyze the prime factorization of the base number:
\[ 6 = 2 \times 3 \]
Therefore, for any natural number $n$, we have:
\[ 6^n = (2 \times 3)^n = 2^n \times 3^n \]
Since the prime factorization of $6^n$ does not contain the prime factor 5, it can never end with the digit 0.

• Let us evaluate the expression for successive natural numbers:
- For $n = 1$: $6^1 = 6$, which ends in 6.
- For $n = 2$: $6^2 = 36$, which ends in 6.
- For $n = 3$: $6^3 = 216$, which ends in 6.
- For $n = 4$: $6^4 = 1296$, which ends in 6.

• We can prove this generally using mathematical induction:
Suppose $6^k$ ends in 6 for some natural number $k$, which means:
\[ 6^k = 10m + 6 \] for some integer $m$.
Now, let us find the value for $n = k + 1$:
\[ 6^{k+1} = 6^k \times 6 = (10m + 6) \times 6 \]
\[ 6^{k+1} = 60m + 36 = 60m + 30 + 6 \]
\[ 6^{k+1} = 10(6m + 3) + 6 \]
This is also in the form $10M + 6$, meaning it ends with the digit 6.
By mathematical induction, $6^n$ ends in 6 for all natural numbers $n$.


Step 4: Final Answer:
The digit with which $6^n$ ends is always 6.
Hence, option (B) is correct.