Am I doing the prove of this problem correctly by using the quotient remainder theorem and division into cases? Please let me know if I did it correctly or not, if not then please show me how to prove this using quotient of theorem and division into cases? Thank you Part a.23.Prove that for any integersa q $ (a?-3)Preof (by contradichon):Suppose not19 E zByThat is suppose a€ Z and 9 (a2-3).9q by definition of divisibilty-quotient remainder the orem,Jk E Z,a? -3thea=qkt|ora - qk +2Case 1832 +1Thena-3 = (42+1)?-3 byCon 18q+1)-3= 89q² +18q-?subsh tutiom{by algebra.2- q (9q?+ a) -2Let k- 942 +aqof integers under addition and multipli caton, Then a-3=9k-2,KEZ SInce 3,9²,q E Z by the closureThusa²-3= 94 = 9K- 2for Some integers q andkresults eesttineat of partawhichcontradi ctstheCase 2a = 99 +2 Thena²-3= (9+2)-3 by subshtehen(An²+36g +4)-3942 +49 ,K tZ sınce 3,4,q²,q EZ by the closureunder additon and nuliplication .Then a²-3=Let k%3Dof integersqK t).qu tl fur some integers9 and KThusa?- 3= 99which contradict the resultsRay

We can solve this by using definition of division algorithm

Part a. 23. Prove that for any integers a 9 la2-3) Preof (by contradichon): Su ppose not 19 e z That is suppose a € R and 9 (a3) by definition of dimisibilty. a? -3 By quotient remainder the orem,Jk E Z, a=qkt| the a - qk +2 or Case 18 a = 32 +1 Then a-3 = (42+1)²_3 by Can +18q11)-3 = 89q² +18q-? - q (9q?+ a) -2 subsh tutiom %3D (by alzebra. 2 Let K- 942 +aq of integers under addition and multipli caton, Then a-3=9k-2, KEZ SInce 3,9²,9 E Z by the closure Thus a?-3 = 9q = 9K-2 for Some integers q andk results eesttanet of parta which contradi cts the Case 2 94 +2 Then a = a²-3= (9+2)-3 by subshtehen (An2+364 +4)-3 ktk - 942 +49.Kt Z since 3,4,q,2 EZ by the closure additon and nualholicahoo

Part a. 23. Prove that for any integers a 9 la2-3) Preof (by contradichon): Su ppose not 19 e z That is suppose a € R and 9 (a3) by definition of dimisibilty. a? -3 By quotient remainder the orem,Jk E Z, a=qkt| the a - qk +2 or Case 18 a = 32 +1 Then a-3 = (42+1)²_3 by Can +18q11)-3 = 89q² +18q-? - q (9q?+ a) -2 subsh tutiom %3D (by alzebra. 2 Let K- 942 +aq of integers under addition and multipli caton, Then a-3=9k-2, KEZ SInce 3,9²,9 E Z by the closure Thus a?-3 = 9q = 9K-2 for Some integers q andk results eesttanet of parta which contradi cts the Case 2 94 +2 Then a = a²-3= (9+2)-3 by subshtehen (An2+364 +4)-3 ktk - 942 +49.Kt Z since 3,4,q,2 EZ by the closure additon and nualholicahoo

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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