Answers

2014-04-01T20:25:39+05:30
A=bq+r
a>b
0 \leq r<b

required remainders are:
0,1,2
a=3q+0.
a=3q+1.
a=3q+2.

1 case:      a=3q
             cubing ;  both ;  sides

a³ = 27q³.
    =9(3q³)
    ==9m
where m = 3q³.

2 case:    a=3q+1
          cubing  ;  both  ;  sides
a³=(3q+1 )³
=( 3q)³ +(1)³+3*3q*1 ( 3q+1)
=27q³+27q²+9q+1
=9( 3q³+3q²+q)+1
==9m+1
where m=3q³+3q²+q

3 case:   a=3q+2
           cubing  ;  both  ;  sides
a³=(3q+2)³
=(3q)³ + (2)³ + 3*3q*2( 3q+2)
=27q³+54q²+36q+8
=9( 3q³+6q²+4q)+8
==9m+8
where m =3q³+6q²+4q

therefore, cubes of any positive integers is either in the form of  9m, 9m+1, 9m+8 for integer 'm'.
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