Number Theory in Mathematics Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report Number theory is a branch of mathematics that studies numbers, particularly whole numbers, and their properties and relationships. It explores patterns, structures, and the behaviors of numbers in different situations. Number theory deals with the following key concepts:Prime Numbers: Properties, distribution, and applications of prime numbers.Divisibility: Rules and relationships of numbers dividing each other.Greatest Common Divisor (GCD) and Least Common Multiple (LCM): Finding common factors and multiples.Modular Arithmetic: Remainders and "clock arithmetic."Number Patterns: Squares, cubes, and other numerical sequences.Congruences: Relationships between numbers in modular systems.Number SystemNumber System explains different kinds of numbers used in mathematics, from natural to complex numbers, and various number systems like binary and decimal.Natural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersImaginary NumbersComplex NumbersThe four common Types of Number Systems are:Binary Number SystemOctal Number SystemDecimal Number SystemHexadecimal Number SystemBasic ConceptsHere, you'll learn foundational number theory concepts like divisors, multiples, prime numbers, HCF, LCM, and modular arithmetic.DivisorsDivisibility RulesSum of DivisorsNumber of DivisorsMultiplesMultiplication TablesHCF and LCMPrime NumbersPrime FactorizationSieve of EratosthenesSegmented SieveGreatest Common Divisor (GCD)Euclidean AlgorithmExtended Euclidean AlgorithmLeast Common Multiple (LCM)Modular ArithmeticModular AdditionModular Multiplication: Modular Multiplicative InverseModular DivisionModular ExponentiationFermat's Little TheoremCo-Prime NumbersEuler’s Totient FunctionAdvanced ConceptsDive into deeper number theory topics such as the Chinese Remainder Theorem, Diophantine equations, and number-theoretic identities.Chinese Remainder TheoremWilson’s TheoremDiophantine EquationsLinear Diophantine EquationsPell’s EquationBezout's IdentityMobius FunctionMobius InversionPrime DistributionUnderstand how prime numbers are spread across the number line and explore interesting patterns and famous conjectures related to them.Distribution of PrimesPrime Number TheoremGoldbach's ConjecturePatterns in PrimesTwin PrimesCousin PrimesPrime TripletsPrime QuadrupletsMiscellaneous Topics of Number TheoryThis section includes special and fascinating types of numbers and principles like Fibonacci numbers, perfect numbers, and the Pigeonhole Principle.Catalan NumbersFibonacci SequenceFarey SequencesPigeonhole PrinciplePerfect NumbersDeficient NumbersAbundant NumbersAmicable NumbersAutomorphic NumbersMagic NumbersTriangular NumbersTetrahedral NumberHexagonal NumbersLucas PrimesHardy-Ramanujan NumbersPrograms for Number TheoryPractice number theory through programming with code examples that solve problems like GCD, LCM, primality testing, and more.Find the GCD of two numberFind the LCM of two numberCalculate the Factorial of a numberBasic and Extended Euclidean algorithmsPrimality Test to check if a number is prime or notPrimality Test to check if a number is prime or not using the Fermat MethodPrimality Test to check if a number is prime or not using Miller–RabinPrimality Test to check if a number is prime or not using Solovay-StrassenNumber Theory BooksThis section lists useful books to help you learn and understand number theory better.Elementary Number Theory by David M. BurtonAn Introduction to the Theory of Numbers by G.H. Hardy and E.M. WrightA Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael RosenAlgebraic Number Theory by Jürgen Neukirch Number Theory Number Theory Number System in Maths How many types of number systems are there? Hexadecimal Number System What are Numbers? Complex Numbers Binary Number System How are complex numbers used in real life? Prime Numbers in Maths Comment More infoAdvertise with us A abhishek1 Follow Improve Article Tags : Engineering Mathematics Maths Tutorials Maths-Categories Math Tutorials +1 More Similar Reads Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. 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