Show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q + 1, where q is some integer.

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.