Question

Show that any positive odd integer is of the form $4q+1$ or $4q+3$ for some integer $q.$

Answer 1

As per Euclid division lemma

if $a$ and $b$ are $2$ positive numbers then

$a=bq+r,where0<r<b$

Let

positive integer be $a$ and $b=4$

Hence

$\begin{array}{c}a=4q+r,where0<r<4\\ hence,r=0,1,2,3\end{array}$

only for $r=1$ and $3$.the integer a is odd.

$then,a=4q+1anda=4q+3isoddno.$