Question

Use euclid's division lemma to show that the cube of any positive integer is of the form 9q or 9q+10 or 9q+8,where q is some integer

Answer 1

Let a be any positive integer and b = 3
a = 3q + r, where q and r are greater than or equal 0 and r < 3

Therefore, every number can be represented as these three forms. There are three cases.

Case 1: When a = 3q,
a3 = 9m
Where m is an integer such that m = 3q3
Case 2: When a = 3q + 1,
a3 = (3q +1)3
a3 = 27q3 + 27q2 + 9q + 1
a3 = 9(3q3 + 3q2 + q) + 1
a3 = 9m + 1
Where m is an integer such that m = (3q3 + 3q2 + q)
Case 3: When a = 3q + 2,
a3 = (3q +2)3
a3 = 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