Investigate integers through divisibility, residues and factorisation. Begin at the first unfamiliar idea; the prerequisite links help you find a shorter route. You can open any lesson.
- N01
Divisibility of integers
Use divisibility as a statement about an integer multiplier.
Inside: Divisibility and integer multiples · Linear combinations · Parity and impossibility proofs
- N02
Euclidean division and remainders
Represent an integer with a valid bounded remainder.
Inside: Quotient and remainder · Negative inputs · Uniqueness of the remainder
- N03
Greatest common divisor and Bézout
Use common divisors and linear combinations to explain the Euclidean algorithm.
Inside: Euclidean algorithm · Bézout coefficients · Gcd and lcm · Coprimality arguments
- N04
Prime numbers
Prove compositeness or an infinitude claim using a clear contradiction or factorisation.
Inside: Primality and composite numbers · Infinitely many primes · Constructing a factorisation · Sophie Germain identity
- N05
Unique prime factorisation
Use prime-exponent parity to justify perfect-power conclusions.
Inside: Unique prime factorisation · Prime exponents · Squares, cubes and coprime products
- N06
Counting and summing divisors
Translate divisor questions into independent choices of prime exponents.
Inside: Counting divisors by exponent choices · Sum and product of divisors · Perfect numbers
- N07
Modular arithmetic and congruences
Choose a useful modulus and justify cancellation or impossibility.
Inside: Congruence notation · Operations and power cycles · Cancellation and inverses · Linear congruences · Modular obstructions
- N08
Complete and reduced residue systems
Use representatives without confusing them with the residue classes.
Inside: Complete residue systems · Reduced residue systems · Multiplication permutes residues
- N09
Number theory theorems: Fermat, Euler, Wilson and CRT
Select a theorem from its hypotheses and explain when it is unavailable.
Inside: Euler’s totient function · Fermat’s little theorem · Euler’s theorem · Wilson’s theorem · Chinese remainder theorem · Binomial coefficients and expansion in congruences · Digit sums in a base · Carmichael function and theorem (advanced)
- N10
Number bases and digit problems
Translate a digit string into a polynomial in the base.
Inside: Place value in arbitrary bases · Base conversion · Digit constraints and divisibility
- N11
Floor and ceiling functions
Use the defining interval to control a floor expression.
Inside: Floor, ceiling and fractional part · Floor identities and inequalities · Counting and factorial exponents
- N12
Diophantine equations by factorisation
Find all integer solutions and prove completeness.
Inside: Linear Diophantine equations · Factorisation and complete factor pairs · Modular impossibility · Bounds and finite cases · Descent and harder equations
These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.