# 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