Question

use euclid' s division lemma to show that the cube of any positive integer is of the form 9m.9m+1 or 9m +8​

Answer 1

REFER TO THE ATTACHMENT

HOPE IT WILL HELP ✌ ✌

Answer 2

Take the common int he cube of 9 and then take common from 9.

Then Using Euclid division lemma:-

a=bq+r ,here 0 is greater than or = to r less the b

ALL remainders are possible from 3 that is:- 0,1,2

All integers are possible:-

a=3q+0=3q

a=3q+1,

a=3q+2.

Now, Cube both sides:-

(i)a^3=(3q)^3

a^3=27q^3

Now take common here

a^3=9(3q^3)

m= 9(3q^3)

(ii) a^3=(3q+1)^3

a^3=a^3+b^3+3a^2b+3ab^2

a^3=3q^3+1^3+3×3q×3q×1+3×3q×1×1

a^3=27q^3+1+27q^2+9q

a^3=27q^3+27q^2+9q+1

Now take common here

a^3=9q(3q^2+3q+3)+1

m= 3q^2+3q+3.

(iii) a^3=(3q+2)^3

a^3=a^3+b^3+3a^2b+3ab^2

a^3=3q^3+2^3+3×3q×3q×2+3×3q×2×2

a^3=27q^3+8+54q^2+36q

a^3=27q^3+54q^2+36q+8

Now take common here

a^3=9q(3q^2+6q+4)+8

m=3q^2+6q+4