Divisor
A 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; in $a / b = q$ (with remainder $r$), $b$ is the divisor, and it is also called a factor of $n$ when it divides $n$ exactly ($r = 0$). Every integer $n$ has at least two divisors, $1$ and $n$ itself, and 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, and the divisor function $d(n) = \sum_{d \mid n, d>0} 1$ is multiplicative with average order $O(\log n)$.
Example
In $24 / 6 = 4$, the divisor is $6$: you are splitting $24$ into groups of $6$. Divisors of $12$ are $1, 2, 3, 4, 6, 12$, while divisors of $7$ are only $1, 7$, making $7$ prime. The sum of all divisors of $n$ (denoted $\sigma(n)$) appears in the study of perfect numbers, $n$ is perfect if $\sigma(n) = 2n$, and 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
The divisor sets the group size: changing the divisor changes the size of each share. 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.