use euclids division lemma to show that the cube of any positive integer is of the form 9m,9m plus 1or 9m plus 8??

check the online solution

aplying euclid division lemma,

a=bq+r

taking b=3

and we know that 0=<r<b

this implies that r = 0,1,2,

therefore case1:

taking r = 0

a = bq+r

a³=[3q]³+0

a³=27q³

a³=9 [ 3q³ ]

a³=9m  ,taking (3q³) = m because it is a constanat.

case ii

taking r = 1

a=3q+1

a³=(3q+1)³

a³=27q³ +1+9q(3q+1)

a³=27q³  +1+27q² +9q

a³=27q³ +27q² +9q+1

a³=9(3q³+3q²+q)+1

a³=9m+1

similarly u can do the third part