Question

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

Answer 1

(i) Let '$a$' be an even positive integer.
Apply division algorithm with a and b, where $b=2$
$a=(2\times q)+r$ where $0\le r<2$
$a=2q+r$ where $r=0$ or $r=1$
since '$a$' is an even positive integer, 2 divides '$a$'.
$\therefore r=0\Rightarrow a=2q+0=2q$
Hence, $a=2q$ when '$a$' is an even positive integer.
(ii) Let '$a$' be an odd positive integer.
apply division algorithm with $a$ and $b$, where $b=2$
$a=(2\times q)+r$ where $0\le r<2$
$a=2q+r$ where $r=0$ or 1
Here $r\ne 0$ ($\because a$ is not even) $\Rightarrow r=1$
$\therefore a=2q+1$
Hence, $a=2q+1$ when '$a$' is an odd positive integer.