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15 integers from 100 with difference divisible by 7

Source: 1997 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

January 27, 2020
number theorydivisible

Problem Statement

Given 100100 integers, is it always possible to choose 1515 of them such that the difference of any two of the chosen numbers is divisible by 77? What is the answer if 1515 is replaced by 1616?
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