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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)

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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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)
Transcribed Image Text: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)
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