a factory, Widget A is produced using 101 components, resulting in a unique finish determined by the rightmost nonzero digit of 1011. The factory plans to increase production to use mare

than 101 components, denoted as n. Find the smallest n > 101 such that the rightmost nonzero digit of nt is the same as that of 1011

To find the smallest \( n > 101 \) such that the rightmost nonzero digit of \( n^t \) (assuming \( t = 1 \) for simplicity) is the same as the rightmost nonzero digit of \( 101^1 \), we need to first determine the rightmost nonzero digit of \( 101 \).

1. **Calculate the Rightmost Nonzero Digit of 101:**
   The rightmost nonzero digit of 101 is 1, as 101 ends in 1.

2. **Determine the Desired Property for \( n \):**
   We want to find the smallest \( n > 101 \) such that the rightmost nonzero digit of \( n \) is 1. For \( n^t \), where \( t = 1 \), this simplifies to \( n \) itself.

3. **Identify the Pattern of Rightmost Nonzero Digits:**
   A number's rightmost nonzero digit can be found by looking at its last few digits and removing any trailing zeros. Since 101 ends in 1, we need \( n \) to also end in 1 to have the same property.

4. **Find the Smallest \( n > 101 \) Ending in 1:**
   The smallest integer greater than 101 that ends in 1 is 111.

Therefore, the smallest \( n > 101 \) such that the rightmost nonzero digit of \( n \) (or \( n^1 \)) is the same as that of 101 is:
\[
n = 111
\]

Thus, the answer is \( \boxed{111} \).

Would you like more details or have any questions?

Here are 8 relative questions for further exploration:
1. What is the significance of the rightmost nonzero digit in other contexts?
2. How do you determine the rightmost nonzero digit for large exponents, like \( n^{10} \)?
3. Can this method be generalized for any \( t \) if not equal to 1?
4. What happens if the factory plans to use different numbers of components, say \( n = 202 \)?
5. How does the method change if we use \( n = 50 \) instead of \( n = 101 \)?
6. Is there a formula for finding the rightmost nonzero digit for \( n^t \) in general?
7. How can the properties of modulo arithmetic simplify finding the rightmost nonzero digit?
8. What are some practical applications of knowing the rightmost nonzero digit in manufacturing?

**Tip:** When dealing with digits and modular arithmetic, it’s often helpful to break down the problem into smaller, more manageable parts to identify patterns.

Explore how to find the smallest n > 101 such that the rightmost nonzero digit of nt matches that of 101. This problem involves number theory and modular arithmetic concepts, useful for advanced high school students.

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