MCQ
Choose the correct answer from the given four options in the following questions:
For some integer q, every odd integer is of the form:
For some integer q, every odd integer is of the form:
- Aq.
- Bq + 1.
- C2q.
- ✓2q + 1.
✓
Answer
Correct option: D.
2q + 1.
We know that, odd integers are 1, 3, 5, ...
So, it can be written in the form of 2q + 1.
where, q = integer = Z
or q = ···, -1, 0, 1, 2, 3, ...
$\therefore$ 2q + 1 = ... -3, -1, 1, 3, 5,
Alternate Answer
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 = 2q + r, where
$0\leq\text{r}<2$
⇒ a = 2q + r, where r = 0 or r = 1
⇒ a = 2q or 2q + 1
when a= 2q + 1 for some integerq, then clearly a is odd.
So, it can be written in the form of 2q + 1.
where, q = integer = Z
or q = ···, -1, 0, 1, 2, 3, ...
$\therefore$ 2q + 1 = ... -3, -1, 1, 3, 5,
Alternate Answer
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 = 2q + r, where
$0\leq\text{r}<2$
⇒ a = 2q + r, where r = 0 or r = 1
⇒ a = 2q or 2q + 1
when a= 2q + 1 for some integerq, then clearly a is odd.
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