Discrete Given any set of 30 integers, must there be two that have the same rema

Question

Discrete

Given any set of 30 integers, must there be two that have the same remainder when they are divided by 25? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Either describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.

Explanation / Answer

In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m > 0, then at least one container must contain more than one item

pigeonhole - remainder = 0,1,2,3,4, 5, ...24 {total 25}

pigeons - any 30 integers

since 30 > 25

then at least one remainder must contain more than one integer

hence there must be two that have same remainder when they are divided by 25.

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