Question

Show that the square of any positive integer is of the form $3m$ or, $3m+1$ for some integer $m$.

Answer 1

Let $a$ be any positive integer. Then, it is of the form $3q$ or $3q+1$ or, $3q+2$

So, we have the following cases:

Case $I$: When $a=3q$
In this case, we have
$a^2=(3q{)}^2=9q^2=3q(3q)=3m$ where $m=3q$

Case $II$: When $n=3q+1$
In this case, we have
$a^2=(3q+1{)}^2=9q^2+6q+1=3q(3q+2)+1=3m+1,$ where $m=q(3q+2)$

Case $III$: When $a=3q+2$
In this case, we have
$a^2=(3q+2{)}^2=9q^2+12q+4=9q^2+12q+3+1$
$3(3q^2+4q+1)+1=3m+1$, where $m=3q^2+4q+1$

Hence $a$ Is of form $3m$ or $3+1$