Question

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

Answer 1

Let $a$ be any positive integer and $b=2.$ Then by Eucli'ds division Lemma there exist integers $q$ and $r$ such that

$a=2q+r$ where $0\le$r$<2$

Now $0\le r<2\Rightarrow 0\le r\le 1\Rightarrow r=0$ or $r=1$

[$\because r$ is an integer ]

$\therefore a=2q$ or $a=2q+1$

If $a=2q$, then $a$ is an even integer.

We know that integer can be either even or odd. Therefore, any odd integer of the form $2q+1$