Using Euclid’s divison lemma -

a = bq + r;

We want to show that any even integer is of the form '2q' and any odd integer is of the form'2q + 1'

So, we take the two integers a and 2.

When we divide a with 2, the possible values of remainder are 0 and 1, which means

a = 2q + 0 or a = 2q + 1

Also, 2q is always an even integer for any integral value of q and so 2q + 1 will always be an odd integer. (∵ 'even integer + 1' is always an odd integer)

∴ when a is a positive even integer it will always be of the form 2q

And when a is positive odd integer it will always be of the form 2q + 1.

∴ for every positive integer it is either odd or even.