Factor
ArithmeticA 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.
Example
Factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$ because each divides $12$ exactly. $5$ is NOT a factor of $12$ because $12 / 5 = 2$ remainder $2$.
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.
Definition
A factor of $n$ is any positive integer $d$ such that $n / d$ is also a positive integer ($d \mid n$ with no remainder). Every integer $n > 1$ has a factorization into primes. The number of factors of $n = p_1^{a_1} \cdot p_2^{a_2} \cdots$ is $(a_1+1)(a_2+1)\cdots$
Example
$60 = 2^2 \times 3 \times 5$. Number of factors $= (2+1)(1+1)(1+1) = 12$. Listing: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$.
Key Insight
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.
Definition
In a commutative ring $R$, $a$ is a factor of $b$ if there exists $c$ in $R$ such that $b = ac$. In $\mathbb{Z}$, the factor concept aligns with the divisibility partial order. Unique factorization domains (UFDs) generalize $\mathbb{Z}$: every element factors uniquely into irreducibles. $\mathbb{Z}[x]$, $\mathbb{Q}[x]$, and $\mathbb{Z}[i]$ are all UFDs.
Example
$\mathbb{Z}[\sqrt{-5}]$ is NOT a UFD: $6 = 2 \times 3 = (1+\sqrt{-5})(1-\sqrt{-5})$, two distinct irreducible factorizations. This failure motivated the development of ideal theory by Kummer and Dedekind.
Key Insight
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.