Divisor ($d\mid n$)

A divisor of an integer $n$ is an integer $d$ satisfying the divisibility relation $d\mid n$. Equivalently, $n=dk$ for some integer $k$. The word positive matters when listing or counting divisors: unless signs are restricted, both $d$ and $-d$ qualify.

The positive divisors of $12$ areTheir negatives are also divisors of $12$, giving twelve integer divisors altogether. The same list of positive divisors works for $-12$. One has only the positive divisor $1$, while zero has every integer as a divisor, including zero under our divisibility convention. A nonzero integer never has zero as a divisor.

For a positive integer $n$, the positive divisors come in pairs $(d,n/d)$ whose product is $n$. For $12$, the pairs are $(1,12)$, $(2,6)$ and $(3,4)$. Each pair contains a divisor at most $\sqrt{n}$, so searching through candidate divisors up to the square root suffices to find all pairs. A pair repeats its own member only when $n$ is a perfect square; for $36$, the middle pair is $(6,6)$. Consequently a positive integer has an odd number of positive divisors exactly when it is a perfect square.

The prime factorization gives another complete description. Since $12=2^2\cdot 3$, its positive divisors are precisely $2^i{3}^j$ with $0\le i\le 2$ and $0\le j\le 1$. These independent choices give $(2+1)(1+1)=6$ positive divisors, the value of the divisor function $\tau (12)$. A prime factor is one particular kind of divisor; $4$, $6$ and $12$ show that divisors need not be prime numbers.

Thinking in pairs connects exact grouping with integer factorization: twelve objects can form three groups of four or four groups of three. Thinking in shared divisors gives the greatest common divisor; for $12$ and $18$, the positive common divisors are $1,2,3,6$, and the largest is $6$.

The positive divisors of a positive integer $n$, ordered by divisibility, form a lattice. The greatest lower bound of $a,b\mid n$ is $\gcd (a,b)$, their greatest common divisor; the least upper bound is $\mathrm{lcm}(a,b)$, their least common multiple. Both are again divisors of $n$.

For example, the positive divisors of $12$ are $1,2,3,4,6,12$. The divisors $4$ and $6$ are incomparable under divisibility, but their greatest lower bound is $2$ and their least upper bound is $12$. The bottom and top elements of the whole lattice are $1$ and $n$, respectively. Under prime factorization, the two operations take the coordinatewise minimum and maximum of the prime exponents.

A positive divisor $d$ of a positive integer $n$ is bi-unitary when the greatest common unitary divisor of $d$ and $n/d$ is $1$. The comparison uses unitary divisors, so it differs from requiring the ordinary greatest common divisor to be one.

For $n=p^e$ with $p$ a prime number and $e\ge 1$, consider $d=p^a$, where $0\le a\le e$. The only possible nontrivial common unitary divisor of $p^a$ and $p^{e-a}$ occurs when $a=e-a>0$. Consequently every divisor is bi-unitary except $p^{e/2}$ when $e$ is even.

For example, the bi-unitary divisors of $64=2^6$ are $1,2,4,16,32,64$; only $8$ is excluded. Its unitary divisors are just $1$ and $64$, showing that these two notions differ substantially. For a general positive integer, apply the prime-power condition independently to each prime factor in its prime factorization.

A divisor $d$ of a positive integer $n$ is unitary when $\gcd (d,n/d)=1$. Each prime power in the prime factorization of $n$ must therefore be included in $d$ completely or omitted completely. Consequently, the sum of the $k$-th powers of unitary divisors is ${\prod }_{p^e\parallel n}(1+p^{ek})$.

A unitary divisor $d$ of a positive integer $n$ satisfies $\gcd (d,n/d)=1$. The greatest common unitary divisor of $a,b$ is the largest integer that is a unitary divisor of both. In their prime factorizations, it includes precisely those whole prime powers whose positive exponents agree in $a$ and $b$.

For $a=12=2^2\cdot 3$ and $b=18=2\cdot 3^2$, it is $1$, although the ordinary greatest common divisor is $6$. For $12$ and $20=2^2\cdot 5$, it is $4$. Requiring unitary divisibility on both inputs is essential: merely being a unitary divisor of their ordinary greatest common divisor is a weaker condition.

A pair of positive integers $(a,b)$ with $ab=n$ for a specified positive integer $n$. Pairing each divisor $a$ with $n/a$ gives $\tau (n)/2$ unordered factor pairs for nonsquare $n$, and $(\tau (n)+1)/2$ for a perfect square, where $\tau (n)$ is the divisor function.

Here a proper divisor of a positive integer $n$ means a positive integer divisor other than $n$ itself. Thus $1$ has no proper divisors, and the proper divisors of $12$ are $1,2,3,4,6$.

A squarefree divisor of an integer $n$ is a divisor $d\mid n$ that is a squarefree integer. It is specified by choosing a subset of the distinct prime factors of $n$.