Question

In ques that is USE EUCLID DIVISION LEMMA TO SHOW THAT THE CUBE OF ANY POSITIVE INTEGER IS OF FORM 9m,9m+1 or 9m+8. Inwhy we use b=3 instead of 9 to solve the problem

Answer 1

Answer:

Let a and b be two positive integers, and a>b

a=(b×q)+r where q and r are positive integers and 

0≤r<b

Let b=3 (If 9 is multiplied by 3 a perfect cube number is obtained) 

a=3q+r where 0≤r<3

(i) if r=0,a=3q    (ii) if r=1,a=3q+1      (iii) if r=2,a=3q+2

Consider, cubes of these

Case (i) a=3q

a3=(3q)3=27q3=9(3q3)=9m           where m=3q3 and   'm' is an integer.

Case (ii) a=3q+1

a3=(3q+1)3                [(a+b)3=a3+b3+3a2b+3ab2]

      =27q3+1+27q2+9q=27q3+27q2+