Some divisions come out clean — 12÷3 gives exactly 4, with nothing left over. Others do not — 13÷3 leaves a remainder of 1. 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.
What is Divisibility?
An integer a divides an integer b when b equals a multiplied by some integer k, with no remainder:
a∣bmeansb=a⋅k for some integer k
The expression 3∣12 is true because 12=3⋅4. The expression 5∣12 is false because no integer k satisfies 12=5⋅k — the closest are 5⋅2=10 and 5⋅3=15, neither of which equals 12.
The expression 4∣20 is true because 20=4⋅5. The expression 7∣30 is false because 30=7⋅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."
Terminology and Notation
The vertical bar in a∣b is the standard notation for divisibility. It is a statement — either true or false — not an operation that produces a number. This distinguishes it from the division symbol ÷, which computes a quotient.
A single relationship supports multiple phrasings, all meaning the same thing. When 3∣12:
3 divides 12. Or: 12 is divisible by 3. Or: 3 is a divisor of 12. Or: 3 is a factor of 12. Or: 12 is a multiple of 3.
Each phrasing shifts the emphasis — "divides" foregrounds 3, "is divisible by" foregrounds 12, "factor" and "multiple" name the roles — but the underlying fact is identical in every case: 12=3⋅4 for an integer 4.
The negation uses a slashed bar: 5∤12 means 5 does not divide 12.
Basic Properties
Several properties follow immediately from the definition and hold for all integers.
Every nonzero integer divides itself: a∣a, because a=a⋅1.
The number 1 divides everything: 1∣a for any a, because a=1⋅a.
Every nonzero integer divides 0: a∣0, because 0=a⋅0. Zero is a multiple of every number.
Zero divides nothing except itself: 0∣b requires b=0⋅k=0, so the only value b can take is 0.
Divisibility is transitive. If a∣b and b∣c, then a∣c. If 3∣12 and 12∣60, then 3∣60.
Divisibility distributes over addition and subtraction. If a∣b and a∣c, then a∣(b+c) and a∣(b−c). Since 4∣20 and 4∣12, it follows that 4∣32 and 4∣8.
Divisibility scales with multiplication. If a∣b, then a∣(b⋅k) for any integer k. Since 3∣12, it follows that 3∣36, 3∣60, 3∣120, and so on.
Divisibility and Remainders
For any integers a and n with n>0, the division algorithm guarantees:
a=n⋅q+r,0≤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∣a. When r=0, the division is inexact and n∤a.
The operation that extracts r directly is modulo. The statement amodn=0 is the computational form of n∣a. The statement amodn=0 is the computational form of n∤a. Divisibility poses the question; modulo computes the answer.
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 or 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≡1(mod9), 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.
Factors and Multiples
Every divisibility relationship names two roles. When a∣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,… — 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 an is also a factor. For 24: the pair (3,8) corresponds to 3⋅8=24, and the pair (4,6) to 4⋅6=24. Searching for factors only up to 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.
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=22⋅3 is assembled from the primes 2 and 3. The number 30=2⋅3⋅5 from three primes.
The first primes are 2,3,5,7,11,13,17,19,23,29,… — 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 full treatment, including primality testing and the Sieve of Eratosthenes, appears on the prime numbers page.
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 23⋅32⋅5. No other combination of primes produces 360. The number 84 factors as 22⋅3⋅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=p1a1⋅p2a2⋯ is (a1+1)(a2+1)⋯ — 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,… and the multiples of 6 are 6,12,18,…. The smallest number in both lists is 12, so lcm(4,6)=12.
The two quantities are linked by a clean identity:
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.
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.
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 amodn=0 and the statement n∣a are two forms of the same fact.
Fractions arise when division is not exact. The expression ba 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.