We are tasked with proving that for any subset $S$ of size $n+1$ where each element is chosen from the set $\{1, 2, \dots, 2n\}$, there exist two distinct elements $a$ and $b$ in $S$ such that $a$ divides $b$.

This problem can be solved using the Pigeonhole Principle. Here's how we can approach it step by step:

Step 1: Group elements by their largest odd divisor

Each element $x \in \{1, 2, \dots, 2n\}$ can be written in the form $x = 2^k \cdot m$, where $m$ is an odd number and $k \geq 0$. The number $m$ is called the largest odd divisor of $x$, because $m$ is odd and dividing $x$ by any higher power of 2 will give a non-integer result.

For example:

• $4 = 2^2 \cdot 1$ (largest odd divisor is 1)
• $12 = 2^2 \cdot 3$ (largest odd divisor is 3)

Step 2: Consider how many odd divisors exist

The odd numbers from 1 to $2n$ form a subset of $\{1, 3, 5, \dots, 2n-1\}$, and their total count is exactly $n$. Hence, the possible largest odd divisors are the odd numbers in $\{1, 3, 5, \dots, 2n-1\}$, and there are $n$ such numbers.

Step 3: Apply the Pigeonhole Principle

Since the subset $S$ has $n+1$ elements and there are only $n$ possible largest odd divisors, by the Pigeonhole Principle, there must be at least two elements in $S$, say $a$ and $b$, which share the same largest odd divisor.

Step 4: Show one divides the other

Suppose $a = 2^k \cdot m$ and $b = 2^l \cdot m$, where $m$ is their common largest odd divisor and $k \neq l$. Without loss of generality, assume $k < l$. Then:

$b = 2^l \cdot m = 2^{l-k} \cdot (2^k \cdot m) = 2^{l-k} \cdot a$

This shows that $a$ divides $b$ because $b = 2^{l-k} \cdot a$.

Conclusion

Thus, for any subset $S$ of size $n+1$ chosen from $\{1, 2, \dots, 2n\}$, there must exist two distinct elements $a$ and $b$ such that $a$ divides $b$.

Would you like further details on any part of this solution? Here are five related questions you might find interesting:

1. Can the Pigeonhole Principle be applied to other problems in number theory?
2. How can this proof be extended to other sets, not just $\{1, 2, \dots, 2n\}$?
3. What are some common applications of divisibility in combinatorics?
4. Can we generalize this problem to larger sets where the size of $S$ is different?
5. What is the significance of the largest odd divisor in number theory?

Tip: When using the Pigeonhole Principle, always focus on identifying a structure that partitions elements into fewer groups than the size of your set!