Question

Class X
Math's Exercise 1.1 Q4.

Answer 1

Let, a be any positive integer and b=3 then by Euclid's division algorithm,
a=3q or 3q + 1 or 3q + 2 where 0 less than equal to r < 3
When a= 3q then a2 = 9q2=3(3q)=3m where m= 3q
When a= 3q + 1 then a2 = 9q2 + 6q + 1= 3(3q2 + 2q) + 1= 3m + 1 where m= 3q2 + 2q
When a=3q + 2 then a2= 9q2+12q+4=3(3q2+4q+1)+1=3m+1where m = 3q2+4q+1
therefore the square of any positive integer is of the 3m or 3m + 1

Answer 2

Let any positive integer be 3q ,3q+1,3q+2,

when n = 3q,

n²=(3q)²=9q²

    =3(3q²)

     =3m (where m=3q²)

when n=3q+1,

   n²= (3q +1)² =9q² +6q+1

       =3(3q²+2q)+1

        =3m+1(where m=3q²+2q)

when n=3q+2,

n²= (3q+2)²  =9q²+12q+4

     = 9q²+12q+3+1

     =3(3q2+4q+1) +1

       =3m +1 (where m=3q2+4q+1)

Hence proved that the square of any positive integer is of the form 3m or 3m+1.