Divisor
ArithmeticA divisor is the number by which another number (the dividend) is divided.
Formula
\text{dividend} / \text{divisor} = \text{quotient}
Definition
The divisor is the number you divide by in a division problem.
Example
In $24 / 6 = 4$, the divisor is $6$. You are splitting $24$ into groups of $6$.
Key Insight
The divisor sets the group size. Changing the divisor changes the size of each share.
Definition
In $a / b = q$ (with remainder $r$), $b$ is the divisor. A divisor of $n$ is also called a factor of $n$ when $b$ divides $n$ exactly ($r = 0$). Every integer $n$ has at least two divisors: $1$ and $n$ itself.
Example
Divisors of $12$: $1, 2, 3, 4, 6, 12$. Divisors of $7$: $1, 7$ (only, making $7$ prime). The number of divisors of $n$ is denoted $d(n)$ or $\tau(n)$.
Key Insight
The sum of all divisors of $n$ (denoted $\sigma(n)$) appears in the study of perfect numbers: $n$ is perfect if $\sigma(n) = 2n$ (equivalently, the sum of proper divisors equals $n$).
Definition
In a commutative ring $R$, $b$ is a divisor of $a$ (written $b \mid a$) if there exists $c$ in $R$ such that $a = bc$. Divisibility defines a partial order on the positive integers. The divisor function $d(n) = \sum_{d \mid n, d>0} 1$ is multiplicative and has average order $O(\log n)$. The Dirichlet series of $d(n)$ is $\zeta(s)^2$.
Example
A perfect number satisfies $\sigma(n) = 2n$. The even perfect numbers correspond exactly to Mersenne primes: $n = 2^{p-1}(2^p - 1)$ where $2^p - 1$ is prime (Euler's theorem).
Key Insight
Divisor sums and the structure of divisors are central to analytic number theory. Understanding divisor behavior led to the development of Dirichlet series and, ultimately, to the proof of the prime number theorem.