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

Let ${}^{\prime }{a}^{\prime }$ 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 $2q+1.$

If ${}^{\prime }{a}^{\prime }$ is of the form $2q,$ then ${}^{\prime }{a}^{\prime }$ is an even integer.

Also, a positive integer can be either even or odd.

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