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.

(N/A) Let $a$ be any positive integer and $b=2$. By Euclid's division lemma,for any positive integer $a$ and divisor $b=2$,there exist unique integers $q$ and $r$ such that $a = bq + r$,where $0 \leq r < b$.
Since $b=2$,the possible values for the remainder $r$ are $0$ and $1$ (i.e.,$0 \leq r < 2$).
Case $1$: If $r=0$,then $a = 2q + 0 = 2q$. Since $2q$ is divisible by $2$,$a$ is an even integer.
Case $2$: If $r=1$,then $a = 2q + 1$. Since $2q$ is even,$2q+1$ is not divisible by $2$,so $a$ is an odd integer.
Thus,every positive even integer is of the form $2q$ and every positive odd integer is of the form $2q+1$.