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Even Number


An even number is an integer of the form n=2k, where k is an integer. The even numbers are therefore ..., -4, -2, 0, 2, 4, 6, 8, 10, ... (OEIS A005843). Since the even numbers are integrally divisible by two, the congruence n=0 (mod 2) holds for even n. An even number n for which n=2 (mod 4) also holds is called a singly even number, while an even number n for which n=0 (mod 4) is called a doubly even number. An integer which is not even is called an odd number.

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 of the even numbers is

 (2x)/((x-1)^2)=2x+4x^2+6x^3+8x^4+....

The product of an even number and an odd number is always even, as can be seen by writing

 (2k)(2l+1)=2[k(2l+1)],

which is divisible by 2 and hence is even.


See also

Connell Sequence, Doubly Even Number, Even Function, Odd Number, Parity, Singly Even Number

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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 A005843/M0985 in "The On-Line Encyclopedia of Integer Sequences."

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Even Number

Cite this as:

Weisstein, Eric W. "Even Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EvenNumber.html

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