Even Number

An even number is an integer of the form , where  is an integer. The even numbers are therefore ..., , , 0, 2, 4, 6, 8, 10, ... (OEIS A005843). Since the even numbers are integrally divisible by two, the congruence  holds for even . An even number  for which  also holds is called a singly even number, while an even number  for which  is called a doubly even number. An integer which is not even is called an odd number.

The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.

The generating function of the even numbers is

The product of an even number and an odd number is always even, as can be seen by writing

which is divisible by 2 and hence is even.

REFERENCES:

Commission on Mathematics of the College Entrance Examination Board. Informal Deduction in Algebra: Properties of Odd and Even Numbers. Princeton, NJ, 1959.

Sloane, N. J. A. Sequence A005843/M0985 in "The On-Line Encyclopedia of Integer Sequences."