Question

Show that the square of an odd positive integer can be of the form $6q+1$ or $6q+3$ for some integer $q$.

Answer 1

It is known that any positive integer can be written in the form of $6m,6m+1,6m+2,6m+3,6m+4,6m+5$ for some integer $m$.
Thus, an odd positive integer can be of the form $6m+1,6m+3,6m+5$
We have, $(6m+1{)}^2=36m^2+12m+1=6(6m^2+2m)+1=6q+1$,

where $q=6(m^2+2m)$ is an integer

Consider $(6m+3{)}^2=36m^2+36m+9=36m^2+36m+6+3=6(6m^2+6m+1)+3=6(6m^2+6m+1)+3=6q+3$,
where $q=6m^2+6m+1$ is an integer

Consider $(6m+5{)}^2=36m^2+60m+25=36m^2+60m+24+1=6(6m^2+10m+4)+1=6q+1$,
where $q=6m^2+10m+4$ is an integer

Thus, the square of an odd positive integer can be of the form $6q+1$ or $6q+3$ for some integer $q$