Question

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

Answer 1

Hope this will help u my frnd

if you like this plz mark as brainlist ans

Answer 2

Let a be any positive integer.

Therefore, by Euclid's Division lemma

a=bq+r

Taking b as 3.

a=3q+r, where 0<_r<_3

Therefore

r=0,1&2

CASE 1:

r=0

a=3q+0

a=3q

a^3=(3q) ^3

a^3=27q^3

a^3=9(3q^3)

a^3=9m. {m=3q^3}

CASE 2:

r=1

a=3q+1

a^3=(3q+1)^3

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

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

a^3=9m+1. (m=3q^3+3q^2+q)

CASE 3:

r=2

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. (m=3q^3+6q^2+4q)

Hence proved