Question

solve this 2 questions ​

Answer 1

Positive integers can be of the form 3m , 3m+1 or 3m+2

[By euclid's division lemma, a=bq+r , where 0≤r<b]

case(i) when x=3m

x^2= (3m)^2

x^2=  9m^2

x^2= 3(3m^2)

[let q=3m^2]

x^2= 3q

case(ii) when x=3m+1

x^2= (3m+1)^2

x^2 = 9m^2 + 6m +1

x^2 = 3(3m^2 +2m) +1

[let q=3m^2 +2m]

x^2 = 3q+1

case(iii) when x= 3m+2

x^2= (3m+2)^2

x^2 = 9m^2 + 12m + 4

x^2 = 9m^2 +12m +3 +1

x^2 = 3(3m^2 +4m + 1) +1

[let q=3m^2 +4m +1]

x^2 = 3q+1

==>square of any positive integer is of the form 3q or 3q+1