Question

use Euclid division Lemma to show that the cube of any positive integer in the form of 9 M or 9 M + 1 or 9M + 8

Answer 1

Let the positive integer be a which when devided by 3 gives q as quotient and r as remainder.

So, according to Euclid division lemma

a=bq+r

a=3q+r

where r=0,1,2

then,
a=3q
or
a=3q+1
or
a=3q+2

⭐case 1:--

a=3q

a³=(3q)³
=27q³
=9m{where m=3q³}

⭐Case 2.........
(used identity (a+b)³=a³+3a²b+3ab²+b³)

a=3q+1

a³=(3q+1)³
=27q³+1+27q²+9q
=9m{where m=3q³+3q²+q}+1
=9m+1

⭐Case 3

a=3q+2

a³=(3q+2)³

=27q³+27q²+36q+8

=9m{where m=3q³+3q²+4q}+8
=9m+8

Hence proved...

@Altaf