. Divisibility is the study of when and why the first case occurs, and the consequences that follow from it. The concept threads through factoring, primes, common divisors, and the internal structure of the integers themselves.

Key Terms

Divisor (Factor)— the integer that divides another exactly

Multiple— the result of multiplying a number by any integer

Prime Number— an integer greater than 1 with no divisors other than 1 and itself

Composite Number— an integer with at least one divisor beyond 1 and itself

Prime Factorization— the unique decomposition into a product of primes

Coprime— two integers sharing no common factor other than 1

Greatest Common Divisor— the largest integer dividing both of two given integers

Least Common Multiple— the smallest positive integer divisible by both of two given integers

See All Arithmetic Definitions →

What is Divisibility?

An integer

a

divides an integer

b

when

b

equals

a

multiplied by some integer

k

, with no remainder:

a \mid b \quad \text{means} \quad b = a \cdot k \;\text{ for some integer } k

The expression

3 \mid 12

is true because

12 = 3 \cdot 4

. The expression

5 \mid 12

is false because no integer

k

satisfies

12 = 5 \cdot k

— the closest are

5 \cdot 2 = 10

and

5 \cdot 3 = 15

, neither of which equals

12

.

The expression

4 \mid 20

is true because

20 = 4 \cdot 5

. The expression

7 \mid 30

is false because

30 = 7 \cdot 4 + 2

— a remainder of

2

survives.

Divisibility is a yes-or-no question. Either

a

fits into

b

a whole number of times, or it does not. There is no "almost divides" or "partially divides."

20 tiles in 4 groups of 5: divisible

Twenty tiles fill four complete rows of five with nothing left over, so 5 divides 20 and 20 = 5 · 4 with k = 4: the definition a | b drawn as a full rectangle. A single leftover tile would break the rectangle and the divisibility with it. Change the number or the divisor and watch the rows fill or fail on the divisibility tiles visualizer.

Divisibility is a statement about rectangles: b tiles can be arranged in rows of exactly a.

Divisibility Notation

Notation

Divisibility Notation

One vertical stroke carries the whole subtree: the bar that states, the slash that denies, and the remainder spelling that computes the same fact. Every symbol here is catalogued among the arithmetic symbols with its LaTeX code, and the mathematical keyboard types them directly. The table below collects the five spoken phrasings of one statement.

\gcd(a,b)

and

\operatorname{lcm}(a,b)

are introduced on their own pages; the congruence triple-bar

\equiv

belongs to modulo.

a \mid b

a divides b

A statement, not an operation:

a \mid b

is true or false — it claims

b = a \cdot k

for some integer

k

and produces no number, unlike

\div

, which computes. What is Divisibility? above sets the definition.

CasesFive spoken phrasings share the one mark — divides, is divisible by, is a divisor of, is a factor of, is a multiple of — each shifting emphasis, none changing the fact; the table below lines them up for

3 \mid 12

Do not confuseDivision's argument order. In

3 \mid 12

the small number leads; in

12 \div 3

it trails — the bar and the division sign read their operands in opposite directions, the subtree's most reliable misreading.

Same glyph elsewhereThe identical stroke is "such that" inside set-builder braces and, doubled, the absolute value fence — position decides the job.

5 \nmid 12

five does not divide twelve

The slashed bar denies:

5 \nmid 12

because no integer

k

gives

12 = 5k

. One stroke through the mark manufactures the negation — the same slash-negation family that builds

\neq

and

\notin

CasesThe negation stays a statement — false claims become true denials; proofs by contradiction lean on it, assuming

a \mid b

and deriving

a \nmid b

Do not confuseA fraction slash. The stroke negates the bar, it does not divide anything —

5 \nmid 12

has no numeric value, while

5/12

is a number.

b \bmod a = 0

b mod a equals zero

The computational spelling of the same fact:

a \mid b

exactly when the remainder vanishes — one statement, two dialects, as Divisibility and Remainders below works out. The

\bmod

operator itself is owned by the modulo page.

CasesThe bridge runs both ways: theory prefers the bar (

3 \mid 12

), computation the remainder test (12 % 3 == 0 in code) — choosing the dialect is choosing the audience.

Do not confuseThe congruence triple-bar.

b \bmod a = 0

is an equation about a computed remainder;

b \equiv 0 \pmod a

states the same thing in congruence dress — interchangeable in content, different in grammar.

All arithmetic symbols →↑ Arithmetic

Phrasing	Example (with 3 ∣ 12)	What it foregrounds
"a divides b"	3 divides 12	the divisor a — the one doing the dividing
"b is divisible by a"	12 is divisible by 3	the dividend b — the perspective of the number being divided
"a is a divisor of b"	3 is a divisor of 12	a's role as one of b's divisors
"a is a factor of b"	3 is a factor of 12	a as a multiplicative building block of b
"b is a multiple of a"	12 is a multiple of 3	b as a product of a and some integer

Basic Properties

Several properties follow immediately from the definition and hold for all integers.

Every nonzero integer divides itself:

a \mid a

, because

a = a \cdot 1

.

The number

1

divides everything:

1 \mid a

for any

a

, because

a = 1 \cdot a

.

Every nonzero integer divides

0

a \mid 0

, because

0 = a \cdot 0

. Zero is a multiple of every number.

Zero divides nothing except itself:

0 \mid b

requires

b = 0 \cdot k = 0

, so the only value

b

can take is

0

.

Divisibility is transitive. If

a \mid b

and

b \mid c

, then

a \mid c

. If

3 \mid 12

and

12 \mid 60

, then

3 \mid 60

.

Divisibility distributes over addition and subtraction. If

a \mid b

and

a \mid c

, then

a \mid (b + c)

and

a \mid (b - c)

. Since

4 \mid 20

and

4 \mid 12

, it follows that

4 \mid 32

and

4 \mid 8

.

Divisibility scales with multiplication. If

a \mid b

, then

a \mid (b \cdot k)

for any integer

k

. Since

3 \mid 12

, it follows that

3 \mid 36

3 \mid 60

3 \mid 120

, and so on.

Property	Statement	Example
Reflexivity	every nonzero integer divides itself: a ∣ a	7 ∣ 7, since 7 = 7 · 1
Universal divisor	1 divides everything: 1 ∣ a for any integer a	1 ∣ 100, since 100 = 1 · 100
Zero as multiple	every nonzero integer divides 0: a ∣ 0	7 ∣ 0, since 0 = 7 · 0
Transitivity	if a ∣ b and b ∣ c, then a ∣ c	3 ∣ 12 and 12 ∣ 60 ⇒ 3 ∣ 60
Distribution	if a ∣ b and a ∣ c, then a ∣ (b + c) and a ∣ (b − c)	4 ∣ 20 and 4 ∣ 12 ⇒ 4 ∣ 32 and 4 ∣ 8
Scaling	if a ∣ b, then a ∣ (b · k) for any integer k	3 ∣ 12 ⇒ 3 ∣ 36, 3 ∣ 60, 3 ∣ 120, …

Divisibility and Remainders

For any integers

a

and

n

with

n > 0

, the division algorithm guarantees:

a = n \cdot q + r, \qquad 0 \leq r < n

The integer

q

is the quotient — how many complete copies of

n

fit into

a

. The integer

r

is the remainder — what is left after those copies are removed.

Divisibility corresponds to the case

r = 0

. When the remainder vanishes, the division is exact and

n \mid a

. When

r \neq 0

, the division is inexact and

n \nmid a

.

The operation that extracts

r

directly is modulo. The statement

a \bmod n = 0

is the computational form of

n \mid a

. The statement

a \bmod n \neq 0

is the computational form of

n \nmid a

. Divisibility poses the question; modulo computes the answer.

23 = 5 · 4 + 3: four full rows and three left over

The four complete rows are the quotient q = 4 and the three tiles that could not start a fifth row are the remainder r = 3, with 0 ≤ 3 < 5 exactly as the division algorithm requires. The remainder is what stands between a number and the next multiple below it. Slide the number up by one tile at a time and watch the remainder cycle on the divisibility tiles visualizer.

Modulo arithmetic, treated on its own page, is the study of this leftover on its own.

Divisibility Rules

Testing whether one number divides another does not always require performing the full division. For certain common divisors, patterns in the decimal digits of a number provide instant answers.

A number is divisible by

2

if its last digit is even. By

5

if its last digit is

0

5

. By

10

if its last digit is

0

. By

3

if the sum of its digits is divisible by

3

. By

9

if the sum of its digits is divisible by

9

.

Each rule exploits the structure of base-

10

place value. The last digit determines divisibility by

2

and

5

because

10

is divisible by both. The digit-sum rule for

9

works because

10 \equiv 1 \pmod{9}

, as explained on the modular arithmetic page.

These shortcuts are covered in full — with rules for

2, 3, 4, 5, 6, 8, 9, 10

, and

11

— on the divisibility rules page.

Learn More

Factors and Multiples

Every divisibility relationship names two roles. When

a \mid b

, the number

a

is a factor (or divisor) of

b

, and

b

is a multiple of

a

.

The factors of a number are finite. The number

24

has exactly eight factors:

1, 2, 3, 4, 6, 8, 12, 24

. Every positive integer has at least two factors —

1

and itself — and most have more.

The multiples of a number are infinite. The multiples of

3

are

3, 6, 9, 12, 15, \ldots

— the list extends without end. Every positive integer generates an infinite sequence of multiples.

Factors come in pairs. If

a

is a factor of

n

, then

\frac{n}{a}

is also a factor. For

24

: the pair

(3, 8)

corresponds to

3 \cdot 8 = 24

, and the pair

(4, 6)

4 \cdot 6 = 24

. Searching for factors only up to

\sqrt{n}

is sufficient — every factor above the square root is already paired with one below it.

The full treatment, including counting formulas and systematic methods, appears on the factors and multiples page.

Learn More

Prime Numbers

A prime number is an integer greater than

1

whose only factors are

1

and itself. The number

7

is prime — no integer other than

1

and

7

divides it. The number

12

is not prime — it has factors

2, 3, 4

, and

6

beyond

1

and

12

.

Primes are the atoms of multiplication. Every integer greater than

1

is either prime or can be built by multiplying primes together. The number

12 = 2^2 \cdot 3

is assembled from the primes

2

and

3

. The number

30 = 2 \cdot 3 \cdot 5

from three primes.

The first primes are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots

— an infinite sequence with no largest member. The number

2

is the only even prime; every other even number is divisible by

2

and therefore composite.

The number

1

is not prime. It has only one factor (itself), while the definition requires exactly two. Excluding

1

is not a technicality — it is necessary to preserve the uniqueness of prime factorization.

The sieve finished: only the primes remain unmarked

Every multiple of 2, 3, 5 and 7 has been struck out in turn, and the numbers still standing are exactly those with no divisor other than 1 and themselves. A prime is what the sieve cannot remove. Run the crossing-out one prime at a time on the Sieve of Eratosthenes visualizer.

Primes are the atoms the next section builds every other integer from.

Prime Factorization

The Fundamental Theorem of Arithmetic guarantees that every integer greater than

1

can be expressed as a product of primes, and that this expression is unique up to the order of the factors.

The number

360

factors as

2^3 \cdot 3^2 \cdot 5

. No other combination of primes produces

360

. The number

84

factors as

2^2 \cdot 3 \cdot 7

. The factorization is a fingerprint — it identifies the number completely.

Prime factorization reveals the divisor structure of a number. The number of factors of

n = p_1^{a_1} \cdot p_2^{a_2} \cdots

(a_1 + 1)(a_2 + 1) \cdots

— a formula that reads the answer directly from the exponents.

Factorization also underpins the computation of GCD and LCM. The GCD takes the minimum exponent of each shared prime; the LCM takes the maximum. These connections are developed on the prime factorization page.

GCD and LCM

Two numbers may share common factors, and the largest of these is their greatest common divisor. The numbers

48

and

36

are both divisible by

1, 2, 3, 4, 6

, and

12

. The largest —

12

— is

\gcd(48, 36)

.

The least common multiple runs in the opposite direction. It is the smallest positive integer that both numbers divide into. For

4

and

6

, the multiples of

4

are

4, 8, 12, 16, \ldots

and the multiples of

6

are

6, 12, 18, \ldots

. The smallest number in both lists is

12

, so

\text{lcm}(4, 6) = 12

.

The two quantities are linked by a clean identity:

a \cdot b = \gcd(a, b) \cdot \text{lcm}(a, b)

For

4

and

6

4 \cdot 6 = 24

, and

\gcd(4,6) \cdot \text{lcm}(4,6) = 2 \cdot 12 = 24

.

Three methods exist for computing the GCD: listing factors, prime factorization, and the Euclidean algorithm — which uses modulo repeatedly to reduce the problem. The full treatment appears on the GCD and LCM pages.

gcd(252, 105) = 21 by repeated division

Each row divides the previous divisor by the previous remainder: 252 = 105 · 2 + 42, then 105 = 42 · 2 + 21, then 42 = 21 · 2 + 0. The last nonzero remainder, 21, is the greatest common divisor, found without factoring either number. Enter any pair and watch the remainders shrink to zero on the Euclidean algorithm visualizer.

The GCD and LCM pages develop each method in full, including the one this picture runs.

Related Concepts

Divisibility connects outward to several areas that approach the same ideas from different angles.

Modulo is the computational counterpart. Where divisibility asks a yes-or-no question — does

n

divide

a

? — modulo computes the remainder that answers it. The statement

a \bmod n = 0

and the statement

n \mid a

are two forms of the same fact.

Fractions arise when division is not exact. The expression

\frac{a}{b}

in lowest terms requires dividing both numerator and denominator by their GCD — a divisibility operation. Simplifying fractions is, at its core, factoring out common divisors.

Modular arithmetic extends divisibility into a full arithmetic system where numbers are classified by their remainders. The divisibility rules for

3

9

, and

11

are consequences of how

10

behaves under modular arithmetic — shortcuts derived from congruence properties of the base of our number system.

Summary: A Map of the Divisibility Subtree

The sections above have surveyed the full territory of divisibility: the definition itself, the terminology variants, the basic properties, the link to remainders and modulo, and the major sub-topics (rules, factors, primes, GCD, LCM). Each sub-topic has its own dedicated page where the details, methods, and worked examples live. The table below collects those pages with a one-line indication of what each covers — use it as a map for what to explore next.

Topic	What you'll find there	Page
Divisibility rules	digit-based shortcuts for testing divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11 without performing the full division	/divisibility/rules
Factors and multiples	listing all factors of a number using pair-search up to √n; counting formulas via prime factorization	/divisibility/factors
Greatest common divisor (GCD)	three methods — listing factors, prime factorization, and the Euclidean algorithm	/divisibility/gcd
Least common multiple (LCM)	the smallest shared multiple; the identity a · b = gcd(a, b) · lcm(a, b)	/divisibility/lcm
Modulo (computational counterpart)	the remainder operation that computes the answer to every divisibility question — and powers the Euclidean algorithm and divisibility rules	/arithmetic/modulo

Divisibility FAQ

What does the notation a | b mean?

The vertical bar a | b is read 'a divides b' and means b = a × k for some integer k. It is a statement (true or false), not an operation, and order matters: 3 | 12 is true but 12 | 3 is false. The negation a ∤ b means 'a does not divide b.' This differs from a ÷ b, which computes a quotient.Read more →

How are GCD and LCM related?

For any two positive integers a and b: a × b = gcd(a, b) × lcm(a, b). For 4 and 6: 4 × 6 = 24, and gcd(4,6) × lcm(4,6) = 2 × 12 = 24. Knowing one allows you to compute the other from the product. See the GCD page for the ways to compute it.Read more →

Why is 1 not considered prime?

A prime must have exactly two distinct factors: 1 and itself. The number 1 has only one factor (itself). Excluding 1 preserves the uniqueness of prime factorization — otherwise 12 could be written as 2² × 3 or 1 × 2² × 3 or 1² × 2² × 3, and so on without end.Read more →