MCQ
Every positive odd integer is of the form $.........$ where $'q\ ’$ is some integer.
- ✓$2q+1$
- B$5q+1$
- C$2q+2$
- D$3q+1$
✓
Answer
Correct option: A.
$2q+1$
Let a be any positive integer and $b = 2$
Then by applying Euclid’s Division Lemma,
we have $, a = 2q + r,$
where $0\leq\text{r}<2$
$\Rightarrow r = 0$ or $1$
$\therefore\text{a }2\text{q}$ or $2\text{q}+1.$
Therefore, it is clear that $a = 2q$
i.e., a is an even integer.
Also, $2q$ and $2q + 1$ are consecutive integers,
therefore $2q + 1$ is an odd integer.
Then by applying Euclid’s Division Lemma,
we have $, a = 2q + r,$
where $0\leq\text{r}<2$
$\Rightarrow r = 0$ or $1$
$\therefore\text{a }2\text{q}$ or $2\text{q}+1.$
Therefore, it is clear that $a = 2q$
i.e., a is an even integer.
Also, $2q$ and $2q + 1$ are consecutive integers,
therefore $2q + 1$ is an odd integer.
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