Question

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

Answer 1

Let a be any positive integer and b=2.Then by Euclid's algorithm, a=2q+r for some integer q≥0 and r=0 or r=1, because 0≤r<2. So, a=2q or a=2q+1

If a is of the form 2q then a is an even integer. Also, a positive integer can be either even or odd. Therefore any positive odd integer is of the form 2q + 1