**Question:**

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

**Solution:**

It is known that any positive integer can be written in the form of $6 m, 6 m+1,6 m+2,6 m+3,6 m+4,6 m+5$ for some integer $m$.

Thus, an odd positive integer can be of the form $6 m+1,6 m+3,6 m+5$.

We have, $(6 m+1)^{2}=36 m^{2}+12 m+1=6\left(6 m^{2}+2 m\right)+1=6 q+1$, where $q=6 m^{2}+2 m$ is an integer

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

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

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

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