PROVE THAT IF 6 INTEGER ARE SELECTED FROM THE SET S={1,2,3,4,5,6,7,8,9,10} THEN PROVE THAT THERE ARE 2 INTEGERS SUCH THAT ONE DIVIDES THE OTHER

To prove that if 6 integers are selected from the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), there are two integers such that one divides the other, we can use the **Pigeonhole Principle**.

### Step-by-Step Proof:

1. **Partition the Set \( S \):**
   Consider the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). We can partition this set into the following subsets:
   \[
   \begin{aligned}
   &\{1, 2, 4, 8\}, \\
   &\{3, 6\}, \\
   &\{5, 10\}, \\
   &\{7\}, \\
   &\{9\}.
   \end{aligned}
   \]
   Each subset contains integers such that every number in the subset is a multiple of the smallest number in the subset. For example, in \(\{1, 2, 4, 8\}\), \(2\), \(4\), and \(8\) are all multiples of \(1\).

2. **Applying the Pigeonhole Principle:**
   There are 5 subsets in the partition described above. According to the Pigeonhole Principle, if we choose more than 5 elements from \( S \), at least two of these elements must come from the same subset.

3. **Conclusion:**
   Since we are choosing 6 integers from \( S \), by the Pigeonhole Principle, at least two of the selected integers must be from the same subset. By construction, in each subset, any two elements have the property that one divides the other. Therefore, among the 6 integers selected, there are at least two integers such that one divides the other.

This completes the proof.

### Do you want more details or have any questions?

### Related Questions:
1. What is the Pigeonhole Principle?
2. Can you prove a similar result for the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \)?
3. How can the Pigeonhole Principle be applied to other mathematical problems?
4. What is an example of a problem where the Pigeonhole Principle is used in combinatorics?
5. What is the general form of the Pigeonhole Principle?
6. Can this concept be extended to non-integer sets?
7. What are other examples of proofs that use partitioning techniques?
8. How does this concept relate to the notion of "divisibility" in number theory?

#### Tip: When solving problems using the Pigeonhole Principle, always consider the structure of the set and how it can be partitioned into smaller groups.

Explore a proof using the Pigeonhole Principle with set S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Understand how to show that selecting 6 integers from this set ensures that at least two integers divide each other. Detailed mathematical solution included.

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