18.3. Divisibility in Arbitrary Subsets

Try these first

1. 1.

   Define a strictly increasing sequence.

   Hint

   Think of numbers placed from smallest to largest.

2. 2.

   What is a subsequence?

   Hint

   Consider how you can pick elements but skip some.

3. 3.

   What defines a strictly decreasing sequence?

   • All terms are equal
   • Each term is less than the next
   • Each term is greater than the next

   Hint

   Visualize a staircase going down.

4. 4.

   True or False: A subsequence must consist of consecutive elements.

   • True
   • False

   Hint

   Think about how you might skip numbers in a list.

5. 5.

   Prove that in any sequence of 10 distinct positive integers, there exists a subsequence of length 4 that is either strictly increasing or strictly decreasing.

   Hint

   Look for the lengths you can compare among backs and see how they lead to pairs.

6. 6.

   Given a set of 15 random numbers, verify if the assertion that you can find a subsequence of length 5 that’s either increasing or decreasing holds true.

   Hint

   Divide the numbers into groups based on parity and analyze their compositions.