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The 61 pages that link to Divisibility, each with the reason it gives.

• Prime numberNarrower topic: A prime is defined by exactly which positive integers divide it.

• Rational numberRelated: Common factors determine whether a fraction can be reduced.

• IntegerRelated: Divisibility organizes integers into factors and multiples.

• Modular arithmeticRelated: Two integers are congruent precisely when their difference is divisible by the modulus.

• Natural numberRelated: Divisibility organizes natural numbers into multiples and underlies factorization.

• Number theoryBroader topic: It organizes integers into factors and multiples, the starting point for many questions in number theory.

• Diophantine equationRelated: Divisibility tests often rule out candidate solutions or constrain their possible values.

• Prime factorizationNarrower topic: A prime factorization makes an integer's divisors easy to characterize.

• Greatest common divisorNarrower topic: The greatest common divisor is defined by which integers divide every input.

• Chinese remainder theoremRelated: Compatibility of noncoprime congruences is expressed through divisibility.

• Fundamental theorem of arithmeticRelated: Prime factors are characterized through divisibility, the relation used to compare factorizations.

• Partial orderRelated: Divisibility partially orders positive integers while leaving many pairs incomparable.

• Partially ordered setBroader topic: On positive integers, divisibility gives a familiar partial order with incomparable pairs.

• Integer factorizationNarrower topic: Factorization depends on identifying integers that divide the target exactly.

• Congruence (number theory)Narrower topic: Two integers are congruent exactly when their difference is divisible by the modulus.

• Coprime integersNarrower topic: Common divisors are the basis of the coprimality test.

• Root of unityRelated: An n-th root is also a d-th root whenever d divides n.

• Transitive relationRelated: If one integer divides a second and the second divides a third, it divides the third.

• Gaussian integerRelated: Gaussian integer factorization is defined through divisibility in ℤ[i].

• Reflexive relationRelated: Every nonzero integer divides itself, giving divisibility a reflexive pattern on nonzero integers.

• Unique factorization domainRelated: Factorization uniqueness is expressed through divisibility and associate factors.

• Euclid's lemmaNarrower topic: The lemma is a rule about how divisibility behaves across a product.

• Trial divisionNarrower topic: Each trial asks whether a candidate divides the target with no remainder.

• Divisor functionNarrower topic: Divisor functions are defined by collecting integers related to n through divisibility.

• IdealRelated: In the integers, ideal membership is exactly divisibility by a generator.

• Sieve of EratosthenesRelated: Crossing out multiples relies on recognizing which numbers each prime divides.

• Strict partial orderRelated: On positive integers, proper divisibility is a strict partial order.

• Bézout's identityNarrower topic: The gcd is defined through common divisors, while the identity also yields divisibility tests.

• Least elementRelated: On positive integers, divisibility orders the set with least element one.

• OneRelated: One divides every integer, making it a universal divisor.

• Antisymmetric relationRelated: On positive integers, mutual divisibility implies equality, making divisibility antisymmetric.

• Hasse diagramRelated: Divisibility among the divisors of an integer produces a classic Hasse diagram.

• Rational root theoremNarrower topic: The theorem states that the root’s numerator and denominator divide specific coefficients.

• Symmetric relationCompared with: Divisibility is generally asymmetric: 2 divides 4, but 4 does not divide 2.

• Factor theoremRelated: Having x − a as a factor means the polynomial is divisible by it.

• Least common multipleNarrower topic: The least common multiple is defined by shared divisibility and minimal positive size.

• Minimal elementRelated: Under divisibility on positive integers above one, the primes are minimal elements.

• p-adic valuationRelated: For integers, v_p(x)≥k exactly when p^k divides x.

• Even numberRelated: An integer is even precisely when 2 divides it.

• Arithmetic functionNarrower topic: Many arithmetic functions depend on which integers divide their input.

• Édouard LucasRelated: Divisibility patterns in recurrence sequences underpin several Lucas results.

• Euclid's theoremRelated: The proof tests whether listed primes divide the constructed number.

• Lowest termsRelated: A fraction is reducible precisely when a number greater than one divides both parts.

• Triangular numberRelated: The product of consecutive integers is always divisible by two, ensuring an integer result.

• Twin primeNarrower topic: Testing divisibility by smaller primes establishes whether a candidate is prime.

• Legendre's formulaNarrower topic: The formula counts multiples of successive powers of p among factorial factors.

• Divisibility ruleNarrower topic: A rule is a shortcut for deciding this exact-multiple relation.

• Extended Euclidean algorithmNarrower topic: The algorithm's remainders preserve the common divisors of each successive pair.

• Erdős–Straus conjectureRelated: Many known constructions split into cases according to divisibility conditions on n.

• Zsigmondy's theoremNarrower topic: The theorem compares which primes divide the nth difference and its predecessors.