Question

Show that every positive even integer is of the form $2q$ and that every positive odd integer is of the form $2q+1$, where q is an integer.

Answer 1

Let $a$ be any positive integer and $b=2$.

Then, By Euclid's algorithm,

$a=2q+r$, for some integer $q\ge 0$, and $r=0$ or $r=1$, because $0\le r<2$.

So, $a=2q$ or $a=2q+1$

If $a$ is of the from $2q$, then $a$ is an even integer. Also, a positive integer can be either even or odd.

Therefore, any positive odd integer is of the form $2q+1$.