Question

Please do both questions. wrong answers will be given thumbs down.

Question 7. Prove using the Division Lemma that Yn E Z, n3 n is divisible by 3 (any proof not using the Division Lemma will receive no credit). Question 8. Define a relation ~ on R \ {0} by saying x ~ y İfzy > 0. (a) Prove that is an equivalence relation (b) Determine all distinct equivalence classes of~ prove that your answer is correct.

Answer 1

Let n^3 - n = n (n^2 - 1) = (n - 1) (n) (n + 1) - - - (1) claim n^3 - n is divisible by 3. dividing n by 3 we get by Division Lemma n = 3q + r 0 lessthanorequalto r < 3 for some q elementof mu Implies r = 0 or 1 or 2 (i) if r = 0 then n = 3q therefore (1) Implies n^3 - n = 3q (3q - 1) (3q + 1). therefore 3 divides R.H.S Implies 3 divides L.H.S Implies 3/n^3 - n (ii) if x = 1 Implies n = 3q + 1 Implies n - 1 - 3q (i) implies n^3 - n = 3q (3q + 1) (3q + 2) also 3 divides R.H.S Implies 3 divides L.H.S Implies 3/n^3 - n (iii) if r = 2 then n = 3q + 2 then (i) Implies n^3 - n = (3q + 1) (3q + 2) (3q + 3) = 3 (3q + 1) (3q + 2) (q + 1) Here also 3 divides R.H.S Implies 3 divides L.H.S Implies 3 divides n^3 - n In