Question

use Euclid's division lemma to show that the cube of any positive integer is of the form 9m,9m+1 or 9m+8
​

Answer 1

Step-by-step explanation:

Let a and b be two positive integers and a greater than b.

a=(b× q)+r

0≤r∠b

Let b=3

a= 3q+r where 0≤r∠3

Case(i):a=3q

a³=(3q)³=23q³=9(3q³)=9m

Case(ii):a=3q+1

a³=(3q+1)³

use formula,

[(a+b)³=a³+b³+3a²b+3ad²]

⇒27q³+1+27q²+9q=27q³+27q²+9q+1

⇒9(3q³+3q²+1)+1

⇒9m+1

Where m=3q³+3q²+q

Case(iii):a=3q+2

a³=(3a+2)³=27q³+8+54a²+26q

⇒27q³-54q²+36q+8=9(3q³+3q²+4q)+8

9m+8, where m=3q³+6q²+4q

∵Cube of any positive integer is either of the form 9m,9m+1 or 9m+8 for some integer m.

Answer 2

Answer:

It is the correct answer.

Step-by-step explanation:

Hope this attachment helps you.