For some integer q,


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

(a) q                           

(b) q +1                  

(c) 2g                       

(d) 2q +1


(d) We know that, odd integers are $1,3,5, \cdots$

So, it can be written in the form of $2 q+1$.

where, $q=$ integer $=Z$

or $\quad q=\cdots,-1,0,1,2,3, \ldots$

$\therefore \quad 2 q+1=\cdots,-3,-1,1,3,5, \ldots$

Alternate Method

Let ' $a$ ' be given positive integer. On dividing ' $a$ ' by 2 , let $q$ be the quotient and $r$ be the remainder. Then, by Euclid's division algorithm, we have

$a=2 q+r$, where $0 \leq r<2$

$\Rightarrow \quad a=2 q+r$, where $r=0$ or $r=1$

$\Rightarrow \quad a=2 q$ or $2 q+1$

when $a=2 q+1$ for some integer $q$, then clearly $a$ is odd.

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