Ask when one integer divides another exactly.
Divisibility is a relation with algebraic consequences, not just a calculation.
Side 146
Number Theory
The arithmetic structure of integers: how divisibility, prime decomposition and modular relationships generate deep patterns from apparently simple whole numbers.
integer→divisibility→prime→congruence→structure
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16working concepts
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Divisibility organizes the integers into arithmetic structure.
Factors, common divisors and prime decomposition create the basic grammar of number theory.
Extract shared arithmetic structure.
The greatest common divisor controls simplification, solvability and modular inverses.
Reduce large divisibility problems recursively.
Repeated remainders find the GCD efficiently and expose linear combinations.
Decompose positive integers into primes.
The fundamental theorem of arithmetic makes prime factorization unique up to order.
Modular arithmetic studies integers through remainders.
Congruence collapses infinitely many integers into finite residue classes while preserving useful arithmetic.
Group integers with the same remainder.
Arithmetic descends consistently to classes modulo a fixed integer.
Solve multiplicative equations modulo n.
An inverse exists exactly when the element is coprime to the modulus.
Combine compatible modular conditions.
Independent congruences can determine a unique solution modulo the product of coprime moduli.
Use multiplicative cycles of residues.
Exponentiation modulo n exhibits regularity tied to group structure.
Diophantine equations ask for integer solutions.
Restricting variables to integers makes many ordinary algebraic equations substantially harder.
Solve ax + by = c in integers.
Solvability is controlled by whether the GCD of a and b divides c.
Parameterize integer right triangles.
A geometric equation becomes a structured family of arithmetic solutions.
Ask whether squares hit a given residue class.
Quadratic reciprocity links residue questions across different primes.
Integer powers create strong constraints.
Growth and factorization often make these equations sparse and difficult.
Number theory is a laboratory for proof techniques.
Simple statements invite arguments by contradiction, descent, induction and construction.
Use contradiction to force a new prime.
Euclid's proof shows how factorization can defeat a finite assumption.
Turn one solution into a smaller impossible one.
Infinite descent excludes certain integer configurations.
Encode arithmetic with functions.
Arithmetic functions such as phi or tau summarize number-theoretic structure.
Study global patterns without simple formulas.
Primes thin out predictably in aggregate even though local gaps remain irregular.
Whole numbers hide highly organized structure. Number theory repeatedly turns elementary statements into problems about decomposition, invariance, residue classes and proof.