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

Step by Step Explanation:
1. Let n be any arbitrary positive integer.
Let us divide n by 2 to get m as the quotient and r as the remainder.
2. Then, by Euclid's division lemma, we have:
n = 2m + r, where 0 > r > 2.
n = 2m or (2m+1), for some integer m.
3. Case 1: When n = 2m
In this case, n is clearly even.
4. Case 2: When n = 2m+1
In this case, n is clearly odd.
5. Thus, for some integer m, every positive even integer is of form 2m and every positive odd integer of the form (2m+1).