Question

Every positive odd integer is of the form 2q+1, where q is some integer.

• A. True
• B. False

Answer 1

The correct option is A

True

Let a be any positive integer and b = 2.
By Euclid's division lemma, there exist integers q and r such that:
a = 2q + r, where 0 $\le$ r < 2
$\Rightarrow$ 0 $\le$ r $\le$ 1
$\Rightarrow$ r = 0 or 1

When r = 0, a = 2q which is completely divisible by 2. Hence it is a positive even integer.

When r = 1, a = 2q + 1 which is gives a remainder 1 when divided by 2. Hence it is an odd integer.

Therefore, every positive odd integer is of the form 2q + 1.