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(15) (+) The set of all positive integers is partitioned into several arithmetic progressions. Show that there is at least one among these arithmetic progressions whose initial term is divisible by its difference.

(15) (+) The set of all positive integers is partitioned into several arithmetic progressions. Show that there is at least one among these arithmetic progressions whose initial term is divisible by its difference.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(15) (+) The set of all positive integers is partitioned into several arithmetic progressions. Show that there is at least one among these arithmetic progressions whose initial term is divisible by its difference. (16) Sixty-five points are given inside a square of side length 1. Prove that there are three of them that span a triangle of area at most 1/32. (17) Let A be an n x n matrix with 0 and 1 entries only. Let us assume that n > 2, and that at least 2n entries are equal to 1. Prove that A contains two entries equal to 1 so that one of them is strictly above and strictly on the right of the other. 15, 17 Pigeon-Hole principle way
Transcribed Image Text:(15) (+) The set of all positive integers is partitioned into several arithmetic progressions. Show that there is at least one among these arithmetic progressions whose initial term is divisible by its difference. (16) Sixty-five points are given inside a square of side length 1. Prove that there are three of them that span a triangle of area at most 1/32. (17) Let A be an n x n matrix with 0 and 1 entries only. Let us assume that n > 2, and that at least 2n entries are equal to 1. Prove that A contains two entries equal to 1 so that one of them is strictly above and strictly on the right of the other. 15, 17 Pigeon-Hole principle way
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