Odd Number
An odd number is any integer that is not divisible by 2, such as 1, 3, 5, 7, and 9.
Formula
n = 2k + 1 \text{ for some integer } k
Definition
An odd number is any whole number that cannot be divided by $2$ evenly, always ending in $1$, $3$, $5$, $7$, or $9$. Formally, an integer $n$ is odd if $n = 2k + 1$ for some integer $k$, equivalently if $n \bmod 2 = 1$; the product of two odd numbers is always odd, while the sum of two odd numbers is always even, and $1$ is odd. Algebraically, an odd integer is any element of the coset $1 + 2\mathbb{Z}$ in $\mathbb{Z}/2\mathbb{Z}$: odd numbers are not closed under addition but are closed under multiplication, and many number-theoretic results (quadratic reciprocity, the Legendre symbol, binary quadratic forms) have separate statements for the prime $2$ versus odd primes.
Example
$7$, $13$, $35$, and $101$ are odd, since $7 / 2 = 3$ remainder $1$. Odd $\times$ Odd = Odd ($3\times5=15$), Odd + Odd = Even ($3+5=8$), Odd $\times$ Even = Even ($3\times4=12$), and the sum of the first $n$ odd numbers equals $n^2$ ($1+3+5+7 = 16 = 4^2$). Fermat's Last Theorem needed separate proofs for $n=4$ (Fermat) and for odd prime exponents (Wiles), reflecting how often mathematics treats the prime $2$ as a special case.
Key Insight
If you try to pair up all the objects and one is always left over, the total is odd. The sum of the first $n$ odd numbers equaling $n^2$ has a beautiful visual proof: each successive odd number adds an L-shaped "gnomon" to a growing square of dots. The special role of $2$ among primes, being the only even one, accounts for a disproportionate share of case-splits and special conditions throughout number theory.