Odd Number
ArithmeticAn 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. Odd numbers always end in 1, 3, 5, 7, or 9.
Example
$7$, $13$, $35$, and $101$ are odd. $7 / 2 = 3$ remainder $1$ (not even).
Key Insight
If you try to pair up all the objects and one is always left over, the total is odd.
Definition
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. The sum of two odd numbers is always even. $1$ is odd.
Example
Odd $\times$ Odd = Odd ($3\times5=15$). Odd + Odd = Even ($3+5=8$). Odd $\times$ Even = Even ($3\times4=12$). Sum of first $n$ odd numbers $= n^2$ ($1+3+5+7 = 16 = 4^2$).
Key Insight
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.
Definition
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. In number theory, many results distinguish odd from even primes: the prime $2$ is the unique "ramified" prime in $\mathbb{Q}$, and many formulas (e.g., for quadratic residues) have separate cases for odd primes.
Example
Fermat's Last Theorem for $n=4$ (Fermat) and for odd prime exponents (Wiles) required separate proofs. The Legendre symbol, quadratic reciprocity, and the theory of binary quadratic forms all have separate statements for the prime $2$ vs. odd primes.
Key Insight
The special role of $2$ among primes stems from it being the only even prime. This single exception accounts for a disproportionate share of case-splits and special conditions throughout number theory.