Question

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. [3 MARKS]

Answer 1

Concept : 1 Mark
Application : 2 Marks

Let a be any positive odd integer and b = 6.

We can apply Euclid's division algorithm on a and b = 6.

$\Rightarrow a=6q+r$

We know that value of b = 6.

$\Rightarrow 0\le r<6$

$\therefore$ All possible values of a are:

a = 6q
a = 6q+1
a = 6q+2
a = 6q+3
a = 6q+4
a = 6q+5

Here 6q, 6q+2 and 6q+4 are divisible by 2 since 2 is factor of them.

$\Rightarrow$ They are not positive odd integers.

$\therefore$ 6q+1, 6q+3 and 6q+5 are positive odd integers.

$\therefore$ Any positive odd integer is of the form (6q+1) or (6q+3) or (6q+5).