Divisibility ($\mid$)

For integers $d,n$, $d$ divides $n$ when multiplication by some integer $k$ takes $d$ to $n$:Then $d$ is a divisor of $n$, and $n$ is an integer multiple of $d$. The notation $d\nmid n$ means that no such integer $k$ exists. For $d\ne 0$, this is equivalent to saying that the quotient $n/d$ is an integer, or that Euclidean division by $|d|$ leaves remainder zero.

For example, $3\mid 12$ because $12=3\cdot 4$, whereas $5\nmid 12$ because $12=5\cdot 2+2$. Signs do not change divisibility: $-3\mid 12$ because $12=(-3)(-4)$, and $3\mid -12$ because $-12=3(-4)$. In general, changing the sign of either input preserves the relation.

The definition here permits zero as either input. Every integer divides zero, since $0=d\cdot 0$. In particular, $0\mid 0$; however, $0\nmid n$ whenever $n\ne 0$, since $0k$ is always zero. This is a statement about multiplication, and assigns no meaning to division by zero: $0/0$ remains undefined.

Two useful properties come directly from the defining multiplication. Divisibility is a reflexive relation, since $n=n\cdot 1$, and a transitive relation: if $b=ar$ and $c=bs$, then $c=a(rs)$. Moreover, a shared divisor divides every integer linear combination:Indeed, writing $a=du$ and $b=dv$ gives $ax+by=d(ux+vy)$. For example, $6\mid 12$ and $6\mid 18$ imply $6\mid (18-12)$. This small observation explains why the Euclidean algorithm preserves common divisors, and why Bézout's identity can certify a greatest common divisor.

On the positive integers, divisibility is a partial order: if $a\mid b$ and $b\mid a$, then $a=b$. The restriction matters, since $3\mid -3$ and $-3\mid 3$ although $3\ne -3$. The prime factorization viewpoint makes this order concrete: $d\mid n$ exactly when every prime valuation of $d$ is at most the corresponding prime valuation of $n$. Thus one relation connects equal grouping, prime factorization, and modular arithmetic.

Book VII, definitions 3 and 5 of Euclid's Elements describe the corresponding positive-number relation through one number measuring another and through integer multiples. The sign and zero conventions above belong to the present integer formulation, rather than to those ancient definitions. ^[ref]

A positive integer $d$ is a common divisor of integers $a,b$ when $d\mid a$ and $d\mid b$. For positive $a,b$, their largest common divisor is their greatest common divisor. The identity $a=qb+r$ implies that $a,b$ and $b,r$ have the same positive common divisors, because $r=a-qb$ and $a=qb+r$.

For a nonnegative integer with decimal expansion $a={\sum }_i{a}_{i}10^i$, the congruence $10\equiv 1(\mathrm{mod}d)$ gives $a\equiv {\sum }_i{a}_i(\mathrm{mod}d)$ whenever $d\mid 9$. Thus divisibility by three or nine is equivalent to divisibility of the sum of its decimal digits by the same number. More generally a base-$b$ expansion has the analogous rule for every divisor of $b-1$. This follows by repeated addition and multiplication in modular arithmetic; it does not require an empirical rule about decimal notation.

A positive integer $n$ lying strictly between consecutive prime squares $p^2<n<q^2$ that is divisible by exactly one of $p,q$. Within a fixed band, the sum is the sum of multiples of $p$ plus the sum of multiples of $q$ minus twice the sum of multiples of $pq$, an inclusion-exclusion calculation using arithmetic progressions.

An integer multiple of an integer $d$ is a product $dk$ with $k$ an integer; equivalently, $d$ is a divisor of that product. The integer multiples of $12$ are $\dots ,-24,-12,0,12,24,\dots$. Zero is an integer multiple of every integer, and the only integer multiple of zero is zero. Shared positive integer multiples lead to the least common multiple.

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 an integer ordered by divisibility form a lattice. The meet is the greatest common divisor, and the join is the least common multiple.

A divisor $d$ of $n$ is bi-unitary when the greatest common unitary divisor of $d$ and $n/d$ equals $1$. For $n=p^e$, all divisors are allowed except $p^{e/2}$ when $e$ is even.

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 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 $n$.

For integers $a,b$ not both zero, their greatest common divisor is the largest positive integer dividing both. It is unchanged by replacing either input by its negative. The convention $\gcd (0,0)=0$ is useful, but this exceptional value is not a largest positive common divisor. For positive inputs, the Euclidean algorithm computes the answer, and Bézout's identity expresses it as an integer linear combination of the inputs.

For $12=2^2\cdot 3$ and $18=2\cdot 3^2$, a common divisor can contain at most one copy of the prime number two and one of the prime number three. Taking the smaller prime valuation at each prime number therefore givesThe check $6=18-12$ is also a Bézout's identity: every common divisor divides this difference, and six itself divides both inputs. More generally, every common divisor divides the greatest common divisor, a stronger statement than merely being no larger than it. The companion least common multiple takes the larger prime valuations, as the diagram there illustrates. For an application, divide the numerator and denominator of the fraction $12/18$ by six to obtain $2/3$; these remaining integers are coprime.

For positive integers $m,n,k$, the greatest common divisor scales by the common multiplier $k$. If $d=\gcd (m,n)$, then $kd$ is a common divisor of $km,kn$. Meanwhile Bézout's identity expresses $kd$ as an integer linear combination of $km,kn$, so their greatest common divisor divides $kd$. These two comparisons prove the identity without prime factorization.

Two integers $a,b$ are coprime when their greatest common divisor is one:For positive integers, this says that the only positive common divisor is one. For signed integers, use $\gcd (a,b)=\gcd (|a|,|b|)$, with absolute values; changing either sign leaves coprimality unchanged. With the convention $\gcd (0,0)=0$, zero is coprime precisely to $1,-1$, because $\gcd (a,0)=|a|$. Both $1,-1$ are coprime to every integer. ^[ref]

The word concerns shared prime factors, not whether either input is a prime number. For instance $8=2^3$ and $9=3^2$ are coprime composite numbers, even though neither is a square-free integer. Conversely, $6=2\cdot 3$ and $10=2\cdot 5$ are square-free integers but are not coprime. Consecutive integers are always coprime: any common divisor of $n,n+1$ divides their difference one. Thus $14,15$ provide another pair of coprime composite numbers.

For positive integers, let $P(n)$ be the prime support of an integer. The equivalent test isThe set intersection of the two prime supports of an integer must be the empty set. Indeed, any positive common divisor greater than one has a prime factor, and any shared prime factor is a common divisor greater than one. Hence coprimality depends on which prime factors occur, rather than on their prime valuations; replacing either input by its radical of an integer preserves coprimality.

For a list of at least two integers, distinguish pairwise coprimality, requiring every pair to be coprime, from joint coprimality, requiring only the greatest common divisor of the entire list to be one. For example,No prime factor belongs to all three prime supports of an integer, yet each pair shares one. By contrast, $4,9,25$ have pairwise coprimality. The distinction is presented by Poritz under the names pairwise and mutually relatively prime. ^[ref]

One use is simplifying fractions. For a fraction $a/b$ with integer entries and $b>0$, divide its numerator and denominator by $d=\gcd (a,b)$ to obtain a reduced fraction with coprime entries. For example, $42/30=7/5$ since $d=6$, and $-42/30=-7/5$ uses the same $d$. A further common divisor would make $d$ too small, so the reduction is complete. In particular, the reduced fraction for zero is $0/1$.

Another use is reversible modular arithmetic: for an integer modulus $m>1$, an integer $a$ has a modular inverse precisely when $a,m$ are coprime. The proof outline uses Bézout's identity: $ua+vm=1$ makes $u$ a modular inverse, and any inverse relation $ua-km=1$ forces every common divisor to divide one. For example, $3\cdot 5-2\cdot 7=1$ gives $5^{-1}\equiv 3(\mathrm{mod}7)$, whereas $6$ has no modular inverse modulo $10$. Thus coprimality is the exact condition allowing division by a residue in modular arithmetic. ^[ref]

A list of integers $a_1,\dots ,a_r$, with $r\ge 2$, has pairwise coprimality when every two entries in distinct positions are coprime. This implies joint coprimality. For positive integers, it means no prime factor occurs in two different entries. For example, $4,9,25$ satisfy this condition.

A list of integers $a_1,\dots ,a_r$, with $r\ge 2$, has joint coprimality when its greatest common divisor is one; define that greatest common divisor by repeated application of the two-input operation. For positive integers, no prime factor may divide every entry. Thus $6,10,15$ have joint coprimality but fail pairwise coprimality. For two entries, the conditions coincide.

For positive integers $a,b$, their least common multiple is the smallest positive integer that is an integer multiple of both. Write it as $\mathrm{lcm}(a,b)$. A positive common integer multiple always exists, since $ab$ is one, but it can be larger than necessary.

For example, the positive integer multiples of $12$ begin $12,24,36,48,60,72$, while those of $18$ begin $18,36,54,72$. Their first positive coincidence is $36$, so $\mathrm{lcm}(12,18)=36$. Both also divide zero, but the word positive excludes zero from this minimum.

Suppose two events happen every twelve and eighteen days, and both happen at day $t=0$. Their next meeting is at day $36$, after three twelve-day intervals and two eighteen-day intervals. Through day $72$, their common event times are $0,36,72$.

The prime factorization explains why this coincidence occurs. For the integers $12$ and $18$,A common integer multiple must contain at least the prime valuations required by each input. Taking the larger prime valuation at each prime number meets both requirements with no extra factors. The greatest common divisor uses the smaller prime valuations instead. This is the same coordinate comparison developed under prime factorization, applied in opposite directions.

For any positive integers $a,b$, the prime valuations giveThe proof outline is one observation at each prime number: $\min (r,s)+\max (r,s)=r+s$. Hence the two sides have identical prime factorizations. Numerically, $6\cdot 36=12\cdot 18=216$. This also gives a calculation using the Euclidean algorithm for the greatest common divisor, without first finding the prime factors of both inputs.

Every positive common integer multiple is an integer multiple of the least common multiple, because its prime valuations must be at least the larger input prime valuations. Including negatives and zero, the common integer multiples of $12$ and $18$ are exactly $36k$ for integers $k$. For more than two positive integers, the same definition and rule of taking the largest prime valuations apply; one may also combine inputs successively. For example, $\mathrm{lcm}(12,18,20)=\mathrm{lcm}(36,20)=180$.

For nonzero signed integers, take absolute values: $\mathrm{lcm}(-12,18)=36$. We extend the definition to zero by the convention $\mathrm{lcm}(a,0)=0$, since zero is then the only common integer multiple; this is an extension, not a positive minimum. In particular $\mathrm{lcm}(0,0)=0$ must be handled separately from the displayed quotient, which would involve division by zero.

The least common multiple is useful for combining fractions with the smallest positive shared denominator. For example,Its scheduling interpretation requires care about starting times. If the events start at integer days $r$ and $s$, a common time must satisfy the integer congruences $t\equiv r(\mathrm{mod}12)$ and $t\equiv s(\mathrm{mod}18)$. The Chinese remainder theorem for noncoprime moduli says a solution exists exactly when $6\mid (r-s)$. If it exists, solutions form one congruence class modulo $36$, but the first meeting need not be day $36$. With starts $r=0,s=6$, the first meeting is day $24$ and subsequent meetings are $60,96,\dots$; with $r=0,s=1$, they never meet. The beauty of the least common multiple is the exact shared spacing, while the integer congruences determine whether and where the spacing begins.

Book VII, Proposition 34 of Euclid's Elements constructs the least common multiple of two numbers, and Proposition 36 extends the construction to three by first combining two of them. In modern notation this is $\mathrm{lcm}(a,b,c)=\mathrm{lcm}(\mathrm{lcm}(a,b),c)$. These are specific ancient treatments of the construction, rather than a claim about its first discovery. ^{[ref] [ref]}