show that the cube of any positive integer is of the form 3m 3m+1 3m+3

Question

show that the cube of any positive integer is of the form 3m 3m+1 3m+3

in progress 0
Liliana 4 weeks 2021-08-16T10:41:25+00:00 1 Answer 0 views 0

Answers ( )

  1. Charlotte
    0
    2021-08-16T10:42:36+00:00

    Answer:

    error: It’s square not cube.

    let ‘ a’ be any positive integer and b = 3.

    we know, a = bq + r , 0 < r< b.

    now, a = 3q + r , 0<r < 3.

    the possibilities of remainder = 0,1 or 2

    Case I – a = 3q

    a² = 9q² .

    = 3 x ( 3q²)

    = 3m (where m = 3q²)

    Case II – a = 3q +1

    a² = ( 3q +1 )²

    = 9q² + 6q +1

    = 3 (3q² +2q ) + 1

    = 3m +1 (where m = 3q² + 2q )

    Case III – a = 3q + 2

    a² = (3q +2 )²

    = 9q² + 12q + 4

    = 9q² +12q + 3 + 1

    = 3 (3q² + 4q + 1 ) + 1

    = 3m + 1 ( where m = 3q² + 4q + 1)

    From all the above cases it is clear that square of any positive integer ( as in this case a² ) is either of the form 3m or 3m +1.

Leave an answer

Browse
Browse

18:9+8+9*3-7:3-1*13 = ? ( )