Question

If any positive even integer is of the form $4q$ or $4q+2$, then $q$ belongs to:

• A. whole number
• B. irrational number
• C. Both A & B
• D. none of these

Answer 1

The correct option is A whole number

Let $a$ be any positive even integer and $b=4$.Then by division algorithm,

$a=4q+r$ for some integer $q\ge 0$ and $r=0,1,2,3$
So,
$a=4q$ or,
$4q+1$,
$4q+2$
$4q+3$
Because $0\ge r\ge 4$
Now,
$4q$ i.e $2\left(2q\right)$ is an even number
$\therefore$ $4q+1$ is an odd number
$4q+2$ i.e. $2(2q+1)$ is an even number
$\therefore (4q+2)+1=4q+3$ is an odd number
Thus, We can say that any positive even integer can be written as in the form of $4q,4q+2$ where $q$ is the whole number