Even Number
An even number is any integer that is exactly divisible by 2, such as 0, 2, 4, 6, and 8.
Formula
n = 2k \text{ for some integer } k
Definition
An even number is any whole number that can be divided by $2$ with nothing left over, always ending in $0$, $2$, $4$, $6$, or $8$. Formally, an integer $n$ is even if $n = 2k$ for some integer $k$, equivalently if $n \bmod 2 = 0$; zero is even, and the sum, difference, and product of two even numbers are always even, as is the product of an even number and any integer. Algebraically, an even integer is any element of the coset $0 + 2\mathbb{Z}$ in $\mathbb{Z}/2\mathbb{Z}$; the parity function $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ (mapping $n$ to $n \bmod 2$) is a ring homomorphism, and the even integers form the ideal $2\mathbb{Z}$ in $\mathbb{Z}$.
Example
$4$, $16$, $28$, and $100$ are even, since $4 / 2 = 2$ leaves no remainder, and any number ending in $4$ (like $3{,}874$) is even. Even + Even = Even ($4+6=10$), Even + Odd = Odd ($4+5=9$), Even $\times$ Even = Even ($4\times6=24$), and Even $\times$ Odd = Even ($4\times5=20$). In the proof that $\sqrt{2}$ is irrational, parity is key: if $p^2$ is even then $p$ is even (since odd squared is odd), a step that relies on the closure of even integers under multiplication.
Key Insight
Even numbers can always be split into two equal groups: if you can pair up all the objects, the total is even. The parity rules (even/odd) are a simple case of arithmetic mod $2$, letting you predict the parity of a result without computing it, a useful check in proofs. Parity is the simplest non-trivial example of a modular equivalence class, and its arithmetic properties illustrate all the key features of modular arithmetic and homomorphisms in a setting students can easily visualize; in characteristic-$2$ fields (like $GF(2)$), the notion of parity collapses entirely, since every element satisfies $n + n = 0$.