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 some integer.

Answer 1

By Euclid's division lemma:

Let us suppose that $a$and $b$ are two positive integers such that $a>b$.

Euclid's division lemma expression:

$a=bq+r,0\le r<b$

Putting $b=2$ (it is a positive even integer)

$a=bq+r,0\le r<2$ $[i.e.,r=0,1]$

If $r=0$,$a=2q$ which is divisible by $2\Rightarrow 2qiseven$

If $r=1$, $a=2q+1$ which is not divisible by $2\Rightarrow 2q+1isodd$

That means every positive integer is either even or odd.

So, if $a$ is an even positive integer then it is of the form $2q$ and if $a$ is an odd positive integer then it is of the form $2q+1$.

Hence, proved.