We are tasked with proving that if one chooses $n + 1$ distinct odd integers from the set $\{1, 2, 3, \dots, 3n\}$, then at least one of these numbers will divide another.

Step-by-Step Proof:

1. Total Number of Odd Integers in $\{1, 2, 3, \dots, 3n\}$:

The total number of odd integers in $\{1, 2, 3, \dots, 3n\}$ is approximately half of the numbers. Specifically, the odd integers in this set are:

$\{1, 3, 5, \dots, 3n - 2, 3n\}$

This gives us exactly $n$ odd integers. These integers are distinct and odd.

2. Pigeonhole Principle and Reducing Odd Integers:

We now focus on a key observation: any odd integer can be written in the form $x = 2^k \cdot m$, where $m$ is odd and $k \geq 0$ is the power of 2 in the factorization. This form essentially separates the odd part $m$ from the powers of 2 that may multiply it.

In other words, every odd integer can be reduced to a unique "reduced odd integer" by dividing it by the highest power of 2 that divides it (i.e., removing any factors of 2 from its binary representation).

3. Equivalence Classes Based on Reduced Odd Integers:

Using this factorization, every odd integer can be grouped with other odd integers that have the same "reduced odd part." For example, $1$, $2 \cdot 1 = 2$, and $4 \cdot 1 = 4$ are in the same equivalence class (they differ only by powers of 2). Similarly, $3$, $6$, $12$, etc., form another equivalence class.

For each odd number, the process of reducing it by dividing out the powers of 2 places it in an equivalence class with other numbers that are multiples of powers of 2 of that reduced number.

4. Number of Equivalence Classes:

Given the set $\{1, 2, \dots, 3n\}$, there are exactly $n$ distinct odd integers. Each odd integer corresponds to a unique reduced odd integer (no two odd integers reduce to the same odd integer when dividing out powers of 2).

Therefore, the number of equivalence classes formed by these reduced odd integers is $n$.

5. Applying the Pigeonhole Principle:

Now, we are tasked with choosing $n + 1$ distinct odd integers from the set $\{1, 2, 3, \dots, 3n\}$.

Since there are only $n$ equivalence classes (one for each distinct reduced odd integer), and we are selecting $n + 1$ odd integers, by the Pigeonhole Principle, at least two of the selected integers must belong to the same equivalence class.

6. Conclusion:

If two odd integers belong to the same equivalence class, one of them is a multiple of the other (since they differ only by a power of 2). Therefore, one of these integers will divide the other.

Thus, we have proven that if one chooses $n + 1$ distinct odd integers from the set $\{1, 2, 3, \dots, 3n\}$, at least one of them must divide another.

Would you like to see any additional details or clarification?

Related Questions:

1. What is the Pigeonhole Principle, and how does it apply to number theory problems?
2. Can the result be generalized to other sets, such as even numbers?
3. How does the concept of equivalence classes apply to other mathematical structures?
4. What are some real-world applications of the Pigeonhole Principle?
5. How would the result change if we extended the set to include more numbers?

Tip:

When tackling number theory problems, always look for underlying structures, such as divisibility patterns or equivalence classes, which can simplify complex relationships.