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

• 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.

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

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

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

• Aliquot sequenceRelated: Every transition depends on which integers divide the current term.

• Lifting-the-exponent lemmaRelated: The lemma’s hypotheses specify which primes divide the bases and their differences or sums.

• Sophie Germain's theoremRelated: The proof forces divisibility conditions incompatible with a nonzero solution.

• Greatest element and least elementRelated: Among positive divisors of a fixed integer, one is least and that integer is greatest.

• Grimm's conjectureRelated: A prime can be assigned to an integer only when it divides that integer.

• If and only ifRelated: Number-theoretic characterizations often state divisibility iff a remainder or factor condition holds.

• Join and meetRelated: For positive integers ordered by divisibility, join and meet are least common multiple and greatest common divisor.

• Midy's theoremRelated: The theorem’s sum is 10ⁿ − 1, a number divisible by 9 and composed of n nines.

• Proof that π is irrationalRelated: Divisibility constraints turn the constructed integral into an integer.

• Von Staudt–Clausen theoremRelated: The condition p−1 | 2n selects the primes in the denominator.