show that the cube of any positive imtegerte or the form am amt) or am

1.	show that the cube of any positive imtegerte or the form am amt) or ama 8 for someinteres m
Answer» Let a be any positive integer and b = 3 a = 3q + r, where q ≥ 0 and 0 ≤ r < 3 ∴ r = 0,1,2 . Therefore, every number can be represented as these three forms. There are three cases. Case 1: When a = 3q, Where m is an integer such that m = Case 2: When a = 3q + 1, a = (3q +1) ³ a = 27q ³+ 27q ² + 9q + 1 a = 9(3q ³ + 3q ² + q) + 1 a = 9m + 1 [ Where m = 3q³ + 3q² + q ) . Case 3: When a = 3q + 2, a = (3q +2) ³ a = 27q³ + 54q² + 36q + 8 a = 9(3q³ + 6q² + 4q) + 8 a = 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. Hence, it is proved .

show that the cube of any positive imtegerte or the form am amt) or ama 8 for someinteres m

Answer»

Let a be any positive integer and b = 3

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

∴ r = 0,1,2 .

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

Case 1: When a = 3q,

Where m is an integer such that m =

Case 2: When a = 3q + 1,

a = (3q +1) ³

a = 27q ³+ 27q ² + 9q + 1

a = 9(3q ³ + 3q ² + q) + 1

a = 9m + 1 [ Where m = 3q³ + 3q² + q ) .

Case 3: When a = 3q + 2,

a = (3q +2) ³

a = 27q³ + 54q² + 36q + 8

a = 9(3q³ + 6q² + 4q) + 8

a = 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.

Hence, it is proved .