Show that the square of any positive integer cannot be of form (Sq + 2) or (5q + 3) for any integer q 

Ans : Let a be any positive integer. Take b=5 as the divisor.

Therefore. a = 5m÷r, r=0,1,2,3,4 

Case-1:a=5m²= 25m²=(5m²)=5q Case-2:a=5m+1=5(5m²+2m)=+1=5q+1 Case-3:=5m+2=a²=(5m²+4m)+4=5q+4

Case-4a:=5m+3= a²-5(5m²+6+1)+4=5q+4

Case-5a:=5m+4=a²-5(5m²+8m+3)+1=5q+1

Hence square of any positive integer can not be of the form (5q + 2) or (5q+3) for any integer q