Answer to Question #301758 in Discrete Mathematics for Ashish

Question

Prove that if 30 dictionaries in a library contains a total of 61,327 pages, then at least one of

the dictionaries must have at least 2045 pages.

Answer 1

Let us prove by contradiction that if 30 dictionaries in a library contains a total of 61,327 pages, then at least one of the dictionaries must have at least 2045 pages.

Suppose that each of 30 dictionaries in a library contain at most $2044$ pages. Then 30 dictionaries in a library contains a total of at most $2044\cdot 30=61,320$ pages. Since $61,320<61,327,$ we get a contradiction. Therefore, at least one of the dictionaries must have at least 2045 pages.