Question

Show that if a is odd and 3a+3b is even then b is odd and ab is odd by

1) Direct Proof

2) Proof by contradiction

Answer 1

therefore Since 3at 3b is even is odd and ,n,EZ(vdegers) az 2n,+ such that 3a+3b 2n (atb) 2n 3 divides 2n 2n3n3 () 3 Some for Claim 2 divides 2 for if odd and then does not divide "3 2n thus nE Z such that 2n3(2n,t) by uçing (1) therefore 72n 6ny+ 3 2n 64 3 (n-2n)= 3 divides 3 2 false 2 ivides hence thus 3 , therefove fro that 32n, 2n2= 3C2n) Such (), we have n3n have Now from ,wa 3(atb) 2 (3) 2n4 atbe a+b is atb 2n (2n,ti) b= 2n, even a 2n,t Now as a 2n,t 2n - b 2n 2. (nyn)

2 (п,-m) -2+ 1 2 (ny-n 2ngt heve ng= n-n,-1 odd b 2n+, Now a F > abs (2n,i) (2nt) 4nns 2n,+ 2"5+ 2(2n,"5 ,+)+ 2 "g+where n2n,5 a b 5 2nt is odd ab 2)Assume to thus the Contyayy that such that even, п, € Z b-2n Nou a is qiven to be that thus PPO h E Z such 2n+ a 3 (2n,t)3 (2n, ) 3a+ 3b Now 6n3 t 6, 6nt 6n,t 2+I 2 (3n,+ 3nt) 2n3+where 3n,+ 3n,l^n, 3a+3bs 2n3+ odd, whth is a Contvadict 3a+3b given assumptton brove ab is ion to th odd Assume to the Now we ab is ,thus а. is that even contrery Such that n,EZ Such that Odd, thus 3 qbz 2n, But 2nt ac

abs 2n Now (2n,t) b 2n, 2n b+b 2n, b- 2, - 2 n, b 2 (n,nb) 2n3 bz 2n ,where 9 Tu-uu proved above that it i's even, but hence contvadietion b is odd have