Question

Q32,33

2 27, b=a2 29. b na 28. b-a' 30. b- (b, a) Find the ged of each pair, where a > b. 2 2 2 32. a-h as h 33. a-b2. a-b4 Express the ged of the given numbers as a linear combi- nation of the numbers. 34. 12, 15, 18 36. 12, 18, 20, 24 ごatb 35. 15, 18, 24 37. 15, 18, 20, 28

Answer 1

(a^2 - b^, a^3 - b^3) [(a - b)(a + b), (a - b)(a^2 + ab + b^2)] = (a - b)[a + b, a^2 + ab + b^2] but gcd (a + b, a^2 + ab + b^2) = 1 gcd (a^2 - b^2, a^3 - b^3) = a - b (a^2 - b^2, a^4 - b^4) = [a^2 - b^2, (a^2)^2 - (b^2)^2] = [a^2 - b^2, (a^2 - b^2) middot (a^2 + b^2)] = (a^2 - b^2)[1, a^2 + b^2] but gcd (1, a^2 + b^2) = 1 gcd (a^2 - b^2, a^4 - b^4) = a^2 - b^2