Factor
A factor is a number that divides evenly into another number, leaving no remainder.
Definition
A factor of a number divides into it evenly with nothing left over; every number has at least two factors, $1$ and itself. Formally, a factor of $n$ is any positive integer $d$ such that $n / d$ is also a positive integer ($d \mid n$ with no remainder), and every integer $n > 1$ has a factorization into primes, with the number of factors of $n = p_1^{a_1} \cdot p_2^{a_2} \cdots$ equal to $(a_1+1)(a_2+1)\cdots$. In a commutative ring $R$, $a$ is a factor of $b$ if there exists $c$ in $R$ such that $b = ac$; unique factorization domains (UFDs) generalize $\mathbb{Z}$, where every element factors uniquely into irreducibles, as in $\mathbb{Z}[x]$, $\mathbb{Q}[x]$, and $\mathbb{Z}[i]$.
Example
Factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$, because each divides $12$ exactly, while $5$ is not a factor since $12 / 5 = 2$ remainder $2$. $60 = 2^2 \times 3 \times 5$, so the number of factors is $(2+1)(1+1)(1+1) = 12$: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$. $\mathbb{Z}[\sqrt{-5}]$ is NOT a UFD: $6 = 2 \times 3 = (1+\sqrt{-5})(1-\sqrt{-5})$, two distinct irreducible factorizations, a failure that motivated the development of ideal theory by Kummer and Dedekind.
Key Insight
Finding factors is like finding all the ways to arrange objects into equal rows: $12$ objects can be arranged as $1\times12$, $2\times6$, or $3\times4$ rows. The formula $(a_1+1)(a_2+1)\cdots$ for counting divisors is a direct consequence of the Fundamental Theorem of Arithmetic and the independence of each prime's exponent. The failure of unique factorization in some rings is not a pathology but a deep structural feature; ideal theory restores uniqueness at the level of ideals, forming the foundation of algebraic number theory.