2) a) Let x be an even integer. Show that x³ + x + 6 is always even. b) In a), we showed that x³ + x + 6 is even whenever x is even. Now extend the proof to include all integers? In other words, can you show that x²+x+6 is always even for every integer x? c) Let x, y be positive integers such that x divides 3y and x divides 4. Show that 2y +1024 is a multiple of x. (Hint: Find all values of x that satisfy x divides 4 and try to reason out from there)

2) a) Let x be an even integer. Show that x³ + x + 6 is always even. b) In a), we showed that x³ + x + 6 is even whenever x is even. Now extend the proof to include all integers? In other words, can you show that x²+x+6 is always even for every integer x? c) Let x, y be positive integers such that x divides 3y and x divides 4. Show that 2y +1024 is a multiple of x. (Hint: Find all values of x that satisfy x divides 4 and try to reason out from there)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 10E: 10. Prove or disprove that the set of all nonzero integers is closed with respect to a. addition...

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