Question

Use euclid's division Lemma to show every positive integer is either even or odd if b= 9

Answer 1

Answer:

Let a and b be two positive integers, and a > b

a = (b × q) + r where q and r are positive integers and

0 ≤ r < b

Let b = 3 (If 9 is multiplied by 3 a perfect cube number is obtained)

a = 3q + r where 0 ≤r <3

(i) if r = 0, a = 3q (ii) if r = 1,a = 3q+1 (iii) if r = 2, a = 3q + 2

Consider, cubes of these

Case (i) a = 3q

a³ = (3q)³ = 27q³ = 9(3q³) = 9m where m = 3q³ and 'm' is an integer.

Case (ii) a 3q + 1

a³ = (3q + 1)³ [(a + b)³ = a³ + b³ + 3a²b + 3ab²

= 27q³ + 1 + 27q² + 9q = 27q³ + 27q² + 9q+1

= 9(3q³ + 3q² + q) + 1 = 9m + 1

where m integer. = 3q³+3q2 + q and 'm' is an

Case (iii) a = 3q + 2

a³ = (3q + 2)³ = 27q³ +8+54q² + 36q

= 27q³ − 54q² + 36q +8 = 9(3q³ +6q² +

4q) + 8

9m +8, where m = 3q³ +6q² + 4q and mis an integer.

...cube of any positive integer is either of the form 9m, 9m + 1 or 9m +8 for some integer m.