Question

For some integers \(q\), every odd integer is of the form:

Answer 1

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 ≤ r < 2
⇒ 0 ≤ r ≤ 1
⇒ 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.