Question

use euclids devision lemma to ahow that the cube of any positive integer ia of the form 9m, 9m+1, 9m+8​

Answer 1

Step-by-step explanation:

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Answer 2

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【【【【【...Answer...】】】】】

➡️Let q be any positive integer .

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 written in the form :-

9m , 9m+1 or 9m +8

◆Now,

(3q)^3 => 27q^3 = 9(3q^3)

= 9m , 【 where m =

3q^3】

(3q+1)^3 = (3q)^3 + 3(3q)^2•1 +

3(3q)•1^2 +1

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

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

= 9m +1 【where m =

3q^3+3q^2+q

( 3q+ 2)^3 = (3q)^3+3(3q)^2•2 +3(3q)•2^2 + 8

= 27q^3 + 54q^2 + 36q + 8

= 9( 3q^3 + 6q^2 + 4q ) +8

= 9m + 8 【where m =

3q^3 + 6q^2 + 4q】

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