Question

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

Answer 1

Consider the given integer $2q,2q+1$

By Euclid's lemma,

$a=bq+r,0\le r<b,a,b,q$ are integers.

Let $b=2$

Applying Euclid’s algorithm, we have:

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

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

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

Now, $a$ positive integer can either be even or odd.

Thus, any positive odd integer.

Is of the form $2q+1.$

Hence, this is the answer.