Choose any 100 whole numbers between 1 and 200 inclusive. It is possible that there is not a single pair of numbers that are relative prime among these 100. For example, if one were to choose all even numbers in this range--there are exactly 100 of them--since each number is divisible by 2, then no two among the 100 will be relatively prime.
However, if one were to choose 101 numbers from the range 1 - 200 inclusive, it is guaranteed that there will be at least one pair of relatively prime numbers. Prove it!