**Problem Statement:**Prove or disprove that given a set of 125 integers, one can choose a subset $ S $ of 16 of them so that the difference of any two integers in $ S $ is divisible by 8.**Explanation:**You have a set of 125 integers and need to identify if there exists a subset of 16 integers where the difference between any two integers is a multiple of 8. This means that every pair of integers in this subset differs by a number that can be evenly divided by 8. Consider exploring approaches such as pigeonhole principle or modular arithmetic to tackle this problem and determine the feasibility of forming such a subset.

Given we have set which contains 125 different numbers. When we divide each number in set by 8 we will get remainder either of 0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 . By pigeonhole principles, Out of 125 number at least ( ( quotient of 125 ÷ 8 ) + 1) = 16 number have same remainder. Hence one can choose these number which have same reminder when dividing by 8 . Therefore difference of any two integer in this collection must divides 8.CONCLUSION: Difference of any two integer in this collection must divides 8.

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Prove or disprove that given a set of 125 integers, one can choose a subset of S of 16 of them so that the difference of any two integers in S is divisible by 8.

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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Question

**Problem Statement:**

Prove or disprove that given a set of 125 integers, one can choose a subset $ S $ of 16 of them so that the difference of any two integers in $ S $ is divisible by 8.

**Explanation:**

You have a set of 125 integers and need to identify if there exists a subset of 16 integers where the difference between any two integers is a multiple of 8. This means that every pair of integers in this subset differs by a number that can be evenly divided by 8.

Consider exploring approaches such as pigeonhole principle or modular arithmetic to tackle this problem and determine the feasibility of forming such a subset.

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