Even number

Basic properties

Evenness is one half of the parity classification of integers (the other half being odd). Key algebraic and arithmetic properties follow directly from the form n = 2k:

• Closure under addition: the sum of two even numbers is even.
• Closure under multiplication: the product of any integer with an even number is even.
• Parity interaction: adding an even number to any integer preserves that integer's parity; adding two odd numbers yields an even number.
• Modular view: an integer is even exactly when it is congruent to 0 modulo 2 (n ≡ 0 mod 2).

Examples and simple tests

Common examples include 0, 2, 4, 6, 8, 10 and their negatives (…, −4, −2). Zero is even because 0 = 2·0. Consecutive even numbers differ by 2. Simple tests for evenness include:

• In base 10, a number is even if its last digit is 0, 2, 4, 6, or 8.
• In binary, a number is even when its least significant bit is 0.
• Algebraically, check whether n/2 is an integer.

Uses and significance

Parity is a basic tool across mathematics and computing. It appears in proofs by parity arguments, in combinatorics to count possibilities, and in algorithms where parity checks help detect simple transmission errors (parity bits). Even numbers simplify many divisibility and symmetry arguments because they factor a 2 out of expressions.

History and related ideas

The notion of even and odd dates back to ancient mathematics; early Greek and other civilizations studied parity as part of number theory and classification. The words and symbols evolved over centuries, but the core idea—divisibility by two—remains the same. Parity extends to other algebraic settings (for example, elements of rings can be called even relative to a chosen factor) and connects to modular arithmetic and group theory.

Distinguishing even from odd is one of the simplest number-theoretic tests and underpins many elementary arguments. For further background on integers and number theory see related introductions.