Let a and b be any two positive integers, such that a > b.
Then, a = bq + r, 0 ≤ r < b …(i) [by Euclid’s division lemma]
On putting b = 2 in Eq. (i), we get
a = 2q + r, 0 ≤ r < 2 …(ii)
r = 0 or 1
When r = 0, then from Eq. (ii), a = 2q, which is divisible by 2
When r = 1, then from Eq. (ii), a = 2q + 1, which is not divisible by 2.
Thus, every positive integer is either of the form 2q or 2q + 1.
That means every positive integer is either even or odd. So, if a is a positive even integer, then a is of the form 2q and if a, is a positive odd integer, then a is of the form 2q + 1.