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

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Solution

$\begin{matrix} (6 q + 1) o r (6 q + 3) q \in I A / C t o d i v i s i o n a o r i t h a m a a n d b a r e p o s i t i v e i n t e g e r s \Rightarrow∴ a = b q + r 0 ⩽ r < b \Rightarrow a = 6 q + r 0 ⩽ r < 6 \Rightarrow r = 0, 1, 2, 3, 4, 5 B e c a u s e o d d p o s i t i v e i n t e g e r a r e c o n s i d e r \end{matrix}$

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