Question

show that the cube of any positive integer is either in the form of 9 M, 9 M + 1 or 9 M + 8 for some integer m​

Answer 1

let the possible positive integer b=3

0<r<3

possible values of r are 0,1,2

so,3m,3m+1,3m+2

now

a=3m

cube in both sides

a3=(3m)3

a=27m

=9(3m(3))=9m

a=(3m+1)

again cube in both sides

a3=(3m+1)3

=(3m)3+(1)3+3*3m*1(3m+1)

=(27m)3+1+(27m)2+9m

=9{(3m)3+(3m)2+1m}+1

=9m+1

similarly again cube

a3=(3m+2)3

=(3m)3+(2)3+3*3m*2(3m+2)

=(27m)3+8+(18m)(3m+2)

={(27m)3+(54m)2+36m}+8

=9{(3m)3+(6m)2+4m}+(2)2

a=9m+2

hence,cube of integers is positive.

note-we use identity of (a+b)3

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