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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