Prove that the square of any positive integer is of form 5q,5q+1,5q+4 for some integer q.
Let x be any positive integer
then, x=5q or x=5q+1 or x=5q+4 for integer x
If x=5q
x2=(5q)2=25q2=5(5q2)=5n
where n=5q2
If x=5q+1
x2=(5q+1)2=25q2+10q+1=5(5q2+2q)+1=5n+1
where n=5q2+2q
If x=5q+4
x2=(5q+4)2=25q2+40q+16=5(5q2+8q+3)+1=5n+1
where n=5q2+8q+3
∴ In each three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.