An odd number is an integer of the form , where  is an integer. The odd numbers are therefore ..., , , 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are called even.

Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence  holds for odd . 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 for the odd 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 A005408/M2400 in "The On-Line Encyclopedia of Integer Sequences."