Even Number
ArithmeticAn 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. Even numbers always end in 0, 2, 4, 6, or 8.
Example
$4$, $16$, $28$, and $100$ are even. $4 / 2 = 2$ (no remainder). Any number ending in $4$ (like $3{,}874$) is even.
Key Insight
Even numbers can always be split into two equal groups. If you can pair up all the objects, the total is even.
Definition
An integer $n$ is even if $n = 2k$ for some integer $k$, equivalently if $n \bmod 2 = 0$. Zero is even. The sum, difference, and product of two even numbers are always even. The product of an even and any integer is even.
Example
Even + Even = Even ($4+6=10$). Even + Odd = Odd ($4+5=9$). Even $\times$ Even = Even ($4\times6=24$). Even $\times$ Odd = Even ($4\times5=20$).
Key Insight
The parity rules (even/odd) are a simple case of arithmetic $\bmod 2$. They let you predict the parity of a result without computing it, a useful check in proofs.
Definition
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. The even integers form an ideal $2\mathbb{Z}$ in $\mathbb{Z}$. In characteristic-$2$ fields (like $GF(2)$), the notion of parity collapses: all elements satisfy $n + n = 0$.
Example
In the proof that $\sqrt{2}$ is irrational, parity is key: if $p^2$ is even then $p$ is even (since $\text{odd}^2 = \text{odd}$), a step that relies on the closure of even integers under multiplication.
Key Insight
Parity is the simplest non-trivial example of a modular equivalence class. Its arithmetic properties illustrate all the key features of modular arithmetic and homomorphisms in a setting students can easily visualize.