Let’s refresh the pigeonhole principle.
THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Discrete Mathematics and its applications by Rosen.
When you divide an integer number by 4, there are only 4 possible reminders: 0, 1, 2, and 3.
Let k=4 be the number of possible reminders (the boxes from the pigeonhole principle), k+1=5 be the group of five integers.
Then, by the pigeonhole principle, among the five integers, there are at least two with the same reminder.
In other words, if 5 numbers are placed in 4 boxes, then at least 1 box contain 2 or more numbers.
Related exercises:
- What is the minimum number of students, each of whom comes from one of the 50 states, who must be enrolled in a university to guarantee that there are at least 100 who come from the same state?
- (a) Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to 9. Is the conclusion in part (a) true if four integers are selected rather than five?
- Show that if there are 30 students in a class, then at least two have last names that begin with the same letter
- Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends
