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