Question

Any positive odd integer is of the form 6q + 1 or ______.

• A. 6q
• B. 6q + 2
• C. 6q + 3
• D. 6q + 5

Answer 1

Answer: (C), (D)

The correct options are
C 6q + 3
D 6q + 5

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, the 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.
So, they are not positive odd integers.

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

Hence, any positive odd integer is of the form (6q+1) or (6q+3) or (6q+5).