To prove that any positive integer $n$ has a multiple whose leftmost digit is 1 without using logarithms, we will use pigeonhole principle combined with the properties of modular arithmetic. Here's a structured proof:

Step-by-step proof:

1. Multiples of $n$: Consider the multiples of $n$: $n, 2n, 3n, \dots$.

2. Focus on the leftmost digit: Our goal is to show that one of these multiples will have a leftmost digit of 1. Instead of directly focusing on the leftmost digit, we will focus on the fractional part of the multiples when divided by powers of 10.

3. Scaling the multiples: Suppose we look at the sequence of multiples of $n$ modulo powers of 10. Specifically, we want to examine the fractional part of $\frac{kn}{10^m}$ for increasing values of $k$ and some large $m$. The key observation is that as we look at more multiples, the fractional parts will start "filling up" all possible values in the interval [0, 1].

4. Pigeonhole Principle: Since there are infinitely many multiples of $n$, and only a finite number of possible "first digits" for any multiple when scaled by powers of 10, by the pigeonhole principle, eventually one of these multiples must start with the digit 1 when written in decimal.

5. Detail on modular properties: The idea is that when we examine the multiples of $n$ modulo $10^m$, the fractional parts of $kn / 10^m$ for sufficiently large $m$ will be evenly distributed between 0 and 1 (due to the density of multiples). Therefore, one of these multiples will be arbitrarily close to a value whose leftmost digit is 1.

6. Conclusion: Hence, by the pigeonhole principle and properties of multiples of $n$, we can always find some multiple of $n$ whose leftmost digit is 1.

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Do you want any more details or have any questions?

Here are some follow-up questions based on this concept:

1. How does the pigeonhole principle apply to modular arithmetic problems?
2. Can this principle be extended to finding multiples with any arbitrary leftmost digit?
3. How would this proof change if we allowed the use of logarithms?
4. What role does the density of multiples play in this proof?
5. Can we generalize this idea to other number bases besides decimal?

Tip: The pigeonhole principle is a powerful tool for proving the existence of certain conditions when working with finite sets!