Question

Show that any positive even integer is of the form $4q$ or $4q+2$, where $q$ is a whole number.

Answer 1

Let $a$ be any positive integer and $b=4$, then by Euclid's algorithm,

$a=4q+r$, for some integer $q\ge 0$ and $r=0,1,2,3$

So, $a=4q$ or $4q+1$ or $4q+2$ or $4q+3$ because $0\le r<4$.

Now, $4q$ that is $(2\times 2q)$ is an even number.

Therefore, $4q+1$ is an odd number.

Now, $4q+2$ that is $2(2q+1)$ which is also an even number.

Therefore, $(4q+2)+1=4q+3$ is an odd number.

Hence, we can say that any even integer can be written in the form of $4q$ or $4q+2$ where $q$ is a whole number.