Verified answer

We know that positive integers are whole numbers.

Also, any integer multiplied by 2 is even.

So every even integer is of the form 2q.

We know that, If 1 is added to these even number, it results in a odd number.

So, every positive odd integer is of the form = 2q+1

Hence, So, every positive odd integer is of the form 2q+1.

[tex]\underline{\underline{\bf{Question : -}}}[/tex]

For some integer q, every positive odd integer is of the form :

[tex]\underline{\underline{\bf{Answer : -}}}[/tex]

Let the positive odd integer be a and b = 2

As we know that :

[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies[/tex][tex]\color{black}\boxed{a = bq + r}[/tex]

If b = 2 Then,

[tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\implies\sf{a = 2q + r}[/tex][tex]\:\:\:\:\:\:\:\:\:\:[/tex] [tex]\sf{[0 < r < b]}[/tex]

Positive value of 2 where,

r = 0 , a = 2q + 0 ; 2q

r = 1 , a = 2q + 1

r = 2 , a = 2q + 2

Hence , any positive odd integer is in the form of 2q, 2q + 1 and 2q + 2

Correct options are (i) and (ii)

[tex]\rule{200pt}{4pt}[/tex]