Question

use division algorithm to show that the cube of any positive integer is the form of 9m,9m+1or9m+8​

Answer 1

Answer:

Refer  to the photos

Step-by-step explanation:

Answer 2

Let q be any positive integers. Then it is of the form 3q, 3q + 1 or 3q + 2.

Now,

We have to prove that the cube of each of these can be rewritten in the form 9m, 9m + 1 or 9m + 8.

( 3q)³ = 27 q³ = 9(3 q³) = 9m [ m = 3q³ ]

Now,

( 3q + 1 )³ = (3q)³ + 3(3q)² × 1 + 3(3q) × 1² + 1

= 27q³ + 27q² + 9q + 1

= 9(3q³ + 3q² + q) + 1

= 9m + 1 [ m = 3q³ + 3q² + q ]

And,

( 3q + 2 ) = (3q)³ + 3(3q)² × 2 + 3(3q) ×2² + 1

= 27q³ + 54q² + 36q + 8

= 9(3q³ + 6q² + 4q) + 8

= 9m + 8 [ m = 3q³ + 6q² + 4q ]