Question

Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8.(can we do this sum like shown in the picture)​

Answer 1

$\underline{\green{\sf{ Question}}}$

Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8.

$\\\\\underline{\green{\sf{Step\ by\ step\ explanation}}}$

Given

• Positive integer is either of the form 9m, 9m + 1 or 9m + 8.

Prove That

• Cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8.

$\underline{\green{\sf{Solution}}}$

$\\\\\text{n=bq+r ~~~~~~~~~~Euclid’s Division Lemma}$

Let b= 3

r= 0,1,2

Case I when r= 0

→n= 3q+0

→n³= (3q)³

→n³= 27q³

→n³= 9(3q³)

$→\text{n³= 9m~~~~~~~~(3q³=m)}$

Case II when r= 1

→n= (3q+1)

→n³= (3q+1)³

→n³= 27q³+1³+27q²+9q

→n³= 9(3q³+3q²+q)+1

$→\text{n³= 9m+1~~~~~~~(3q³+3q²+q=m)}$

Case III when r=2

→n= (3q+2)

→n³= (3q+2)³

→n³=27q³+54q²+36q+8

→n³= 9(3q³+6q²+4q)+8

$→\text{n³= 9m+1~~~~~~~(3q³+6q²+4q=m)}$

• Hence the Cube of positive integer is either of the form 9m, 9m + 1 or 9m + 8.