Question

Let S be a set of ten integers chosen from 1 through 50. Show that the set contains at least 2 different (but not necessarily disjoint):

A) Prime numbers
B) Even numbers
C) Odd numbers
D) Multiples of 5

Answer 1

Using the Pigeonhole Principle, we can show that a set of ten integers chosen from 1 through 50 contains at least 2 different prime numbers, even numbers, odd numbers, and multiples of 5.

Step-by-step explanation:

To show that the set contains at least 2 different prime numbers, even numbers, odd numbers, and multiples of 5, we can use the Pigeonhole Principle. Since there are 10 integers chosen from a set of 50 numbers, there must be at least one repetition. Let's consider each case:

1. Prime numbers: There are 15 prime numbers between 1 and 50. If none of the 10 integers chosen are prime numbers, then there would be at most 15 - 10 = 5 prime numbers left. So, there must be at least 2 different prime numbers in the set.
2. Even numbers: There are 25 even numbers between 1 and 50. If none of the 10 integers chosen are even numbers, then there would be at most 25 - 10 = 15 even numbers left. So, there must be at least 2 different even numbers in the set.
3. Odd numbers: Similarly, there are 25 odd numbers between 1 and 50. If none of the 10 integers chosen are odd numbers, then there would be at most 25 - 10 = 15 odd numbers left. So, there must be at least 2 different odd numbers in the set.
4. Multiples of 5: There are 10 multiples of 5 between 1 and 50. If none of the 10 integers chosen are multiples of 5, then there would be at most 10 - 10 = 0 multiples of 5 left. So, there must be at least 2 different multiples of 5 in the set.