18.3.3. Conclusion on Divisibility

Try these first

1. 1.

   Define a strictly increasing sequence. Give an example.

   Hint

   Think about numbers that follow one another and keep getting larger.

2. 2.

   What is a subsequence?

   Hint

   Remember, the order should remain the same.

3. 3.

   What defines a strictly increasing sequence?

   • Each number is greater than the last
   • Each number is less than the last
   • Numbers can repeat

   Hint

   Think about how numbers compare to each other.

4. 4.

   True or False: In every set of n + 1 distinct real numbers, there must be a strictly increasing subsequence of length n + 1.

   • True
   • False

   Hint

   Reflect on the properties of increasing subsequences.

5. 5.

   Using 9 distinct integers from 1 to 20, demonstrate through a sequence how the pigeonhole principle ensures either an increasing or decreasing subsequence of length 5.

   Hint

   Look for how many distinct lengths can be achieved and overlap them.

6. 6.

   Given the sequence of prime numbers up to 30, find a subsequence of at least 4 numbers that is strictly increasing and explain your reasoning.

   Hint

   Remember that primes inherently don't repeat, aiding your subsequence choice.