Mathematics / core
Arithmetic from divisibility to congruences
Understand integer structure using Euclidean algorithms, prime factorization, modular equations, and reciprocity.
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Module 01 core
Arithmetic from divisibility to congruences
Understand integer structure using Euclidean algorithms, prime factorization, modular equations, and reciprocity.
- 01
Divisibility and the Euclidean algorithm
Find greatest common divisors by preserving common divisors while shrinking a remainder problem.
28 min - 02
Bézout identities and integer equations
Use a gcd certificate to decide solvability and parameterize every integer solution of a linear equation.
28 min - 03
Unique prime factorization
Prove that positive integers have one prime structure, then use exponents to compute divisibility information.
28 min - 04
Modular arithmetic and invertible residues
Compute with congruence classes while recognizing exactly when cancellation and division are valid.
28 min - 05
The Chinese remainder theorem
Combine compatible residue constraints and prove both construction and uniqueness of the resulting class.
28 min - 06
Fermat, Euler, and multiplicative orders
Use the finite cycle of an invertible residue to reduce powers while distinguishing a period from the smallest period.
28 min - 07
Primitive roots and cyclic unit groups
Recognize when one residue generates every unit and certify a primitive root by testing prime divisors of the group size.
28 min - 08
Quadratic residues and Legendre symbols
Decide whether a nonzero residue is a square modulo an odd prime using pair counting, Euler’s criterion, and multiplicativity.
28 min - 09
Quadratic reciprocity and its applications
Reduce square-solvability questions between odd primes while tracking the sign and the two supplementary laws.
28 min - 10
Continued fractions and rational approximation
Extract rational convergents from Euclidean divisions and use determinant identities to control their approximation error.
32 min