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

#### Solution

According to Euclid's division lemma, if a and b are two positive integers such that a is greater than b; then these two integers can be expressed as

a = bq + r; where 0 ≤ r < b

Now consider

b = 2; then a = bq + r will reduce to

a = 2q + r; where 0 ≤ r < 2,

i.e., r = 0 or r = 1

If r = 0, a = 2q + r ⇒ a = 2q

i.e., a is even

and, if r = 1, a = 2q + r ⇒ a = 2q + 1

i.e., a is add; as if the integer is not even; it will be odd.

Since, a is taken to be any positive integer so it is applicable to the every positive integer that when it can be expressed as

a = 2q

∴ **a** is even and when it can expressed as

a = 2q + 1; a is odd.

**Hence the required result.**