Course

Book 1. Introduction to Olympiad Number Theory

Book 1. Introduction to Olympiad Number Theory

  • 1. Divisibility and Prime Factorisation
  • 2. GCD, LCM and Euclidean Algorithm
  • 3. Modular Arithmetic I: Residues and Contradictions
  • 4. Modular Arithmetic II: Linear Congruences and Systems
  • 5. Diophantine Equations I: Factorisation and Bounds
  • 6. Infinite Descent I
  • 7. Fermat, Euler and Power Cycles
  • 8. Chinese Remainder Theorem
  • 9. Divisor Counting
  • 10. Digits, Bases and Periodicity
  • 11. Mixed Problems I
  • 12. Mock Olympiads I
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Theory Examples Practice Ladders

Chapters

Chapters

Chapter

Divisibility and Prime Factorisation

The module introduces the precise language of divisibility, primes, factorisation, divisor counting, and first olympiad techniques: consecutive integers, parameters, and factorial constructions.
24 Problems
Theory Examples Practice

Chapter

GCD, LCM and Euclidean Algorithm

The module turns computational GCD into an olympiad tool: Euclid's algorithm, linear combinations, GCDs of expressions, the GCD-LCM relation, coprimality, and numbers of the form \(a^m-1\).
24 Problems
Theory Examples Practice

Chapter

Modular Arithmetic

Remainders, congruences, arithmetic modulo m, power cycles, and modular contradiction.

0 Problems
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Chapter

Modular Arithmetic I: Residues and Contradictions

The module teaches residues as a tool for proving impossibility: tables of squares and cubes, choosing a modulus, last digits, and first modular contradictions.
24 Problems
Theory Examples Practice

Chapter

Congruences and Remainders

Residue classes, linear congruences, compatible remainders, and olympiad-style remainder arguments.

0 Problems
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Chapter

Modular Arithmetic II: Linear Congruences and Systems

The module teaches solving linear congruences, using inverses, knowing when division is allowed, checking system compatibility, and constructing numbers with prescribed remainders.
24 Problems
Theory Examples Practice

Chapter

Diophantine Equations

Integer solutions, linear Diophantine equations, factorisation, modular obstructions, and positivity constraints.

0 Problems
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Chapter

Diophantine Equations I: Factorisation and Bounds

The module teaches first olympiad-style Diophantine equations: linear equations, completing a product, difference of squares, reciprocal equations, modular obstructions, and bounds.
24 Problems
Theory Examples Practice

Chapter

Infinite Descent

Infinite descent, parity lemmas, minimal counterexamples, primitive solutions, and contradiction by smaller solutions.

0 Problems
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Chapter

Infinite Descent I

The module introduces infinite descent: minimal counterexamples, descent by parity and prime divisor, irrationality of square roots, modular obstructions, primitive solutions, and a first Vieta-descent preview.
24 Problems
Theory Examples Practice

Chapter

Fermat and Euler

Fermat's Little Theorem, Euler's totient function, Euler's theorem, modular inverses, and large power computations.

0 Problems
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Chapter

Fermat, Euler and Power Cycles

The module teaches how to work with large powers modulo an integer: short cycles, Fermat's little theorem, Euler's theorem, order of an element, modular inverses, and first restrictions on prime divisors of power expressions.
24 Problems
Theory Examples Practice

Chapter

Chinese Remainder Theorem

The module develops CRT as a method for solving systems of congruences and as an olympiad construction tool: compatibility, non-coprime moduli, shifts, blocks of composite numbers, and constructions with prescribed divisors.
24 Problems
Theory Examples Practice

Chapter

Divisor Counting and Special Numbers

Counting divisors with prime factorization, squarefree divisors, products of divisors, factorial exponents, and special number patterns.

0 Problems
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Chapter

Divisor Counting

The module develops the divisor-counting formula, odd divisors, the square criterion, inverse problems for \( au(n)\), products of divisors, and smallest numbers with prescribed divisor count.
24 Problems
Theory Examples Practice

Chapter

Number Bases

Base conversion, arithmetic in non-decimal bases, binary notation, digit counts, and trailing zeros in other bases.

0 Problems
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Chapter

Digits, Bases and Periodicity

The module translates digit problems into congruences: divisibility rules, last digits, base-\(b\) notation, repunits, decimal periods, and first constructions of numbers with restricted digits.
24 Problems
Theory Examples Practice

Chapter

Fractions, Decimals, and Periodicity

Terminating decimals, repeating decimals, decimal periods, and fraction-to-decimal structure.

0 Problems
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Chapter

Mixed Problems I

The module trains method selection without an announced topic: divisibility, GCD, congruences, factorisation, Diophantine equations, periods, CRT, descent, and constructions.
24 Problems
Theory Examples Practice

Chapter

Divisibility Rules

Digit rules for divisibility, modular proofs of rules, missing digit problems, and last-digit arguments.

0 Problems
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Chapter

Mock Olympiads I

The final module of Number Theory Book 1: training problems without announced methods, imitating short olympiad rounds and consolidating strategy selection.
24 Problems
Theory Examples Practice