The set of even numbers can be written:
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's 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 × 10^{14}, but still no general 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.
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The following laws can be verified using the properties of divisibility and the fact that 2 is a prime number:
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: