Question

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

• A. $q$
• B. $q+1$
• C. $2q$
• D. $2q+1$

Answer 1

The correct option is D $2q+1$

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

$a=2q+r,\text{where}0\le r<2\Rightarrow a=2q+r,\text{where}r=0\text{or}r=1\Rightarrow a=2q\text{or}2q+1\text{when}a=2q+1\text{for some integer}q,{\text{then clearly}}^{\prime }{a}^{\prime }\text{is odd.}$