Question

please answer fast. ​

Answer 1

let 'a' be any positive integer

using euclid's division algoritm

a=bq+r

here b=3 and r=0,1,2

case 1

r=0

a=3q

cubing both sides

a^3 = 27q^3

a^3 = 9(3q^3)

a^3 = 9m ( for m=3q^3 is any integer)

case2

r=1

a=3q+1

cubing both sides

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

a^3 = 27q^3 +1 +3× 3q(3q+1)

a^3 = 27q^3 +9q(3q+1) +1

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

a^3 = 9m +1 ( for m= 3q^3 +q(3q+1) is any integer)

case 3

r=2

a=3q+2

cubing both sides

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

a^3 = 27q^3+ 8 +3×3q×2×(3q+2)

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

a^3 = 9m +8 ( for m= 3q^3 +2(3q+2) is any integer)

Ttherefore cube of any positive integer can be written in the form of 9m , 9m+1 and 9m +8.