Can 6n End with the Digit 0 for Any Natural Number n?
Key Insight
A number ends with the digit 0 if and only if it is divisible by 10. Since 10 = 2 × 5, the number must have both 2 and 5 as prime factors.
Step-by-Step Explanation
Step 1: Prime Factorization of 6
First, express 6 in terms of its prime factors:
6 = 2 × 3
Step 2: Prime Factorization of 6n
Raise 6 to the power of n:
6n = (2 × 3)n = 2n × 3n
Step 3: Check for Required Prime Factors
For 6n to end with 0, its prime factorization must include both 2 and 5:
- 6n has 2n and 3n but no factor of 5 for any natural number n.
Step 4: Conclusion
Since 6n lacks the prime factor 5, it can never be divisible by 10. Thus, 6n cannot end with the digit 0 for any natural number n.
Verification with Examples
Let's test for small values of n:
- 61 = 6 → Ends with 6
- 62 = 36 → Ends with 6
- 63 = 216 → Ends with 6
- 64 = 1296 → Ends with 6
Observation: No matter how large n is, 6n always ends with 6, not 0.
General Rule
A number kn can end with 0 only if the prime factorization of k includes both 2 and 5.
- Example: 10n ends with 0 because 10 = 2 × 5
- Counterexample: 6n cannot end with 0 because 6 = 2 × 3 (missing 5)
Final Answer
No, 6n cannot end with the digit 0 for any natural number n.
Teaching Tip: Ask students to test this logic with other numbers (e.g., 5n, 12n) to reinforce understanding! 😊