# Even and odd numbers

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In mathematics, any integer 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 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 numeral 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, a number expressed in the binary numeral system 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 in the ring of integers, but the odd numbers do not. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.

All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.

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

The Feit-Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.

In wind instruments which are cylindrical and in effect closed at one end, such as the clarinet at the mouthpiece, the harmonics produced are odd multiples of the fundamental frequency. (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason, the harmonics are even multiples of the same frequency, but this is the same as being all multiples of double the frequency and is usually perceived as such.) See harmonic series (music).

## Arithmetic on even and odd numbers

The following laws can be verified using the properties of divisibility and the fact that 2 is a prime number:

### Addition and subtraction

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

### Multiplication

• even * even = even
• even * odd = even
• odd * odd = odd

### Division

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, it is even iff the numerator has more factors of two than the denominator. This will be correct it you add them up right!

## Parity in mathematics

Parity is evenness or oddness of an integer. To say whether a number is even or odd is to specify its parity.

The parity of a permutation (as defined in abstract algebra) is the parity (even or odd) of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions).

It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. See the article on even and odd permutations for an elaboration.