Question

Prove that the square of any positive integer is of form $5q,5q+1,5q+4$ for some integer $q$.

Answer 1

Let x be any positive integer

then, $x=5q$ or $x=5q+1$ or $x=5q+4$ for integer $x$

If $x=5q$

$x^2=(5q{)}^2=25q^2=5(5q^2)=5n$

where $n=5q^2$

If $x=5q+1$

$x^2=(5q+1{)}^2=25q^2+10q+1=5(5q^2+2q)+1=5n+1$

where $n=5q^2+2q$

If $x=5q+4$

$x^2=(5q+4{)}^2=25q^2+40q+16=5(5q^2+8q+3)+1=5n+1$

where $n=5q^2+8q+3$

$\therefore$ In each three cases $x^2$ is either of the form $5q$ or $5q+1$ or $5q+4$ and for integer q.