Question

Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. [3 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Proof : 1 Mark

Let us start with taking $a$, where $a$ is a positive odd integer.

We apply the division algorithm with $a$ and $b=4$

Since $0\le r<4$ , the possible remainders are 0, 1, 2 and 3.

$\Rightarrow a$ can be $4q,4q+1,4q+2,4q+3$ where $q$ is the quotient.

However, since $a$ is odd, $a$ cannot be $4q,4q+2$ (since they are both divisible by 2).

$\therefore$ Any odd integer is of the form $4q+1,4q+3$