Question

Show that every positive even integer is of the form q and that every positive odd integer is of the form 2q + 1, where q is a whole number.

Answer 1

If every positive odd integer is of the form $2q+1$, then the even integer should be $2q$ but here it is given $q$.

Let a be any positive integer and $b=2$. Then, by Euclid’s division there exist integers $q$ and $r$ such that

$A=2q+r$, where $0\le r\le 2$

Now, $0\le r\le 2\to r=0$ or $r=1$

.˙. $a=2q$ or, $a=2q+1$

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

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