Question

use euclid division leema to show yhat the cube of any positive integer is of the form 9m,9m+1,9m+8​

Answer 1

Answer:-

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.

Proof:-

There are three case:-

Case 1:

When a = 3q,

Where m is an integer such that

=> m = (3q)³ = 27q³

=>9(3q³) = 9m

Case 2:

When a = 3q + 1,

=>a ³ = (3q +1) ³

=>a ³ = 27q ³ + 27q² + 9q + 1

=>a ³ = 9(3q ³ + 3q ² + q) + 1

=>a ³ = 9m + 1

=>Where m is an integer such that m = (3q ³ + 3q ²+ q)

Case 3:

When a = 3q + 2,

=>a ³ = (3q +2) ³

=>a ³ = 27q ³ + 54q ² + 36q + 8

=>a ³ = 9(3q ³ + 6q ² + 4q) + 8

=>a ³ = 9m + 8

Where m is an integer such that m = (3q ³ + 6q ²+ 4q)

=>Therefore, the cube of any positive integer is of the form 9m, 9m + 1,

or 9m + 8

Hence, Solved.

Answer 2

hope this help you a lot