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In mathematics an even integer, that is, a number that is divisible by 2, is called **evenly even** or **doubly even** if it is a multiple of 4, and **oddly even** or **singly even** if it is not. (The former names are traditional ones, derived from the ancient Greek; the latter have become common in recent decades.)

These names reflect a basic concept in number theory, the **2-order** of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization.
A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd.
A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even.

The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others.

The ancient Greek terms "even-times-even" and "even-times-odd" were given various inequivalent definitions by Euclid and later writers such as Nicomachus. Today, there is a standard development of the concepts. The 2-order or 2-adic order is simply a special case of the *p*-adic order at a general prime number *p*; see *p*-adic number for more on this broad area of mathematics. Many of the following definitions generalize directly to other primes.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Singly_and_doubly_even

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