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

Explanation

Let a be any positive integer .
Let `" " ` b =2
By Euclid's algorithm
a = 2q + r
for some `q ge0` and `0 le r lt 2` i.e., r = 0 `" "` or `" " r=1`
`therefore " " a = 2q " "` or `" "` 2q +1
If a = 2q , then a is an even integer .
If a = 2q +1 , then a is an odd integer .
Also ,a positive integer is either even or odd.
`therefore` every positive even integer is of the form 2q and every positive odd integer is of the form 2q+1 . `" " ` Hence proved.