. Check whether 8" ends with zero or not. Give reason

To determine whether \(8^n\) (where \(n\) is a positive integer) ends with zero, we need to check if it is divisible by 10. A number ends with zero if and only if it is divisible by 10. For a number to be divisible by 10, it must be divisible by both 2 and 5.

Now, consider \(8^n\):
- \(8 = 2^3\), so \(8^n = (2^3)^n = 2^{3n}\).
- \(2^{3n}\) is clearly divisible by 2, but it does not have any factors of 5.

Since \(8^n\) does not contain the factor 5, it cannot be divisible by 10. Therefore, \(8^n\) **does not end with zero** for any positive integer \(n\).

Would you like more details on this concept or have any questions?

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Here are some related questions:

1. What are the conditions for a number to end in a different digit, like 5 or 6?
2. How do you determine the last digit of a power like \(7^n\)?
3. What are some examples of numbers that always end with a specific digit?
4. How does the prime factorization of a number affect its divisibility by 10?
5. Can any power of a prime number end with zero?

**Tip:** A number ends in zero if it is divisible by both 2 and 5, meaning it must have at least one factor of 10.

Explore whether powers of 8 (8^n) end with zero through detailed mathematical reasoning. Learn about exponentiation, divisibility rules, and prime factorization in determining the last digit of powers. Suitable for advanced high school students and beyond.

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