Question

Mr to show that the cube of any positive integer is either of the form 3m 3m + 1 or 3 m + 2

Answer 1

Let a be any positive integer and b = 3

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

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

Case 1: When a = 3q

a 3=27q3

a3 =3(9q3)

a3=3m

Where m is an integer such that m =

Case 2: When a = 3q + 1,

a 3 = (3q +1) 3

a 3 = 27q 3 + 27q 2 + 9q + 1

a 3 = 3(9q 3 + 9q 2 + 3q) + 1

a 3 = 3m + 1

Where m is an integer such that m =

Case 2: When a = 3q + 2,

a 3 = (3q +2) 3

a 3 =27q3 + 8+18q(3q+2)

a3 = 27q3 +3+w+54q2 + 36q