Question

show that any positive odd integer is of the form $4q+1or4q+3,where$ q is some positive integer.

Answer 1

We have to prove that any positive odd integer is of the form $4q+1$ or $4q+3$, where $9$ is some integer.

We can say by Euclid's division lemma, if $a$ and $b$ are $2$ positive integers, then

$a=bq+r$

where $0\le r<b$

if we consider positive integer be a and b as $4$

Hence $a=4q+r$

Here r is an integer greater than or equal to $0$ and less than $4$, hence r can be either $0,1,2$ or $3$

If $r=1$ then $a=4q+1$

If $r=3$ then $a=4q+3$

This will always be an odd integer.