3) Suppose A is a set containing n+1 integers. Show that it is always possible to find twointegers a and b in A so that a-b is divisible by n. (HINT: Think about partitioningthe integers into classes modulo n.)

This problem can be solved using the Pigeonhole Principle.Theorem: Given a set A containing n + 1…

Answered: 3) Suppose A is a set containing n+1…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 13E

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Similar questions

19. Let and be nonzero integers. Prove that if and only if divides .
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
20. If and are nonzero integers and is the least common multiple of and prove that.
Prove that if and are integers such that and , then either or .
Let a,b,c and d be integers such that ab and cd. Prove that acbd.
Let , and be three nonzero integers. Use definition 2.11 as a pattern to define a greatest common divisor of , and . Use Theorem 2.12 and its proof as a pattern to prove the existence of a greatest common divisor of , and . If is the greatest common divisor of , and , show that . Prove . Definition 2.11: Greatest common Divisor An integer is a greatest common divisor of and if all these conditions are satisfied: is a positive integer. and . and imply . Theorem 2.12: Greatest Common divisor Let and be integers, at least one of them not . Then there exists a unique greatest common divisor of and . Moreover, can be written as for integers and , and is the smallest positive integer that can be written in this form.
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3) Suppose A is a set containing n+1 integers. Show that it is always possible to find two integers a and b in A so that a-b is divisible by n. (HINT: Think about partitioning the integers into classes modulo n.)