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Even and odd numbers

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In mathematics, any integer (whole number) is either even or odd. If it is a multiple of two, it is an even number; otherwise, it is an odd number. Examples of even numbers are -4, 8, 0, and 70. Examples of odd numbers are -5, 1, and 71. The number zero is considered to be even, because it is equal to two multiplied by zero.

The set of even numbers can be written:

Evens = 2Z = {..., -6, -4, -2, 0, 2, 4, 6, ...}.

The set of odd numbers can be shown like this:

Odds = 2Z + 1 = {..., -5, -3, -1, 1, 3, 5, ...}.

A number expressed in the decimal number system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it's odd; otherwise it's even. The same idea will work using any even base. In particular, in the binary numeral system, the number is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is even or odd according to the sum of its digits.

The even numbers form an ideal 2Z in the ring of integers, but the odd numbers do not. An integer is odd if it is congruent to 1 modulo this ideal, in other words if it's congruent to 1 modulo 2, and even if it is congruent to 0 modulo 2.

Goldbach's conjecture, conceived by scientist Christian Goldbach, states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have proven this conjecture to be true for integers up to at least 4 × 1014, but still no proof has been found.

In wind instruments which are cylindrical and closed at one end, such as the clarinet, the harmonics produced are odd multiples of the fundamental.

Table of contents

Arithmetic on even and odd numbers

The following laws follow arithmetic in the factor ring Z/2Z.

Addition and subtraction

  • even ± even = even;
  • even ± odd = odd;
  • odd ± odd = even.


  • even × even = even;
  • even × odd = even;
  • odd × odd = odd.


The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which isn't even or odd, since the concepts even and odd apply only to integers. But when the quotient is an integer:

  • even / odd = even;
  • odd / odd = odd;
  • odd / even is never an integer;
  • even / even could be either.

See also: Even permutation, Parity

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