Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m +1 or 9m +8.

Ans. let a be any positive integer and b =3
a = 3q +r, where q ≥ 0 and 0 ≤ r < 3 Since a =3q or 3q + 1 or 3q +2 Therefore, every number can be represented as these three forms. There are three cases. Case 1 – when a = 3q, a³ = (3q)³ = 27q³ = 9 (3q³)= 9m where m is an integer such that m = 3q³ Case 2 – when a = 3q + 1, a³ = (3q + 1)³ a³ = 27q³ + 27q ²+ 9q + 1 a³ = 9 (3q³ + 3q² + q) + 1 a ³= 9m + 1 where m is integer such that m = (3q³ + 3q² + q) Case 3 – where a = 3q + 2 a³ = (3q + 2) ³ a ³= 27q³ + 54q² + 36q + 8 a³ = 9(3q³+ 6q² + 4q) + 8 a³ = 9m + 8 where m is an integer such that m = (3q³ + 6q² + 4q) therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.³³³ Ans. let a be any positive integer and b =3 a = 3q +r, where q ≥ 0 and 0 ≤ r < 3 Since a =3q or 3q + 1 or 3q +2 Therefore, every number can be represented as these three forms. There are three cases. Case 1 – when a = 3q, a³ = (3q)³ = 27q³ = 9 (3q³)= 9m where m is an integer such that m = 3q³ Case 2 – when a = 3q + 1, a³ = (3q + 1)³ a³ = 27q³ + 27q ²+ 9q + 1 a³ = 9 (3q³ + 3q² + q) + 1 a ³= 9m + 1 where m is integer such that m = (3q³ + 3q² + q) Case 3 – where a = 3q + 2 a³ = (3q + 2) ³ a ³= 27q³ + 54q² + 36q + 8 a³ = 9(3q³+ 6q² + 4q) + 8 a³ = 9m + 8 where m is an integer such that m = (3q³ + 6q² + 4q) therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.³³³

You may also like...

error: Content is protected !!