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

Explanation

Let n be an arbitrary positive integer.
On dividings n by 2, let m be the quotient and r be the remainder.
Then, by Euclid's divison lemma, we have
` n = 2m+r , " where" 0 le r lt 2`
n = 2m or ( 2m+1) ,for some integre m.
Case I when n = 2m.
In the case , n is clearly even.
Case II when n = 2m +1
In the case, n is clearly odd.
Hence, every positive ever integer is of the form 2m and every positive odd integer is of the form ( 2m +1) for some integer m.