Question

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.

Answer 1

Let p be any positive even integer. By definition of even numbers, p is divisible by 2 and the quotient is an integer.
Thus p/2 = some q, where q is an integer. This is equivalent to saying p = 2q. Thus any positive even integeris of the form 2q where q is some integer.
Now we also know that consecutive integers are even and odd alternately. Therefore if p is an even integer, p+1 will be an odd integer.
Since we have shown p is of the form p = 2q, it follows that an odd integer is of the form 2q+1, where q is some integer.