Number Theory (Interesting Facts and Algorithms) Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report Questions based on various concepts of number theory and different types of number are quite frequently asked in programming contests. In this article, we discuss some famous facts and algorithms:Interesting Facts of Number Theory :1. All 4 digit palindromic numbers are divisible by 11.2. If we repeat a three-digit number twice, to form a six-digit number. The result will be divisible by 7, 11 and 13, and dividing by all three will give your original three-digit number.3. A number of form 2N has exactly N+1 divisors. For example 4 has 3 divisors, 1, 2 and 4.4. To calculate sum of factors of a number, we can find the number of prime factors and their exponents. Let p1, p2, ... pk be prime factors of n. Let a1, a2, .. ak be highest powers of p1, p2, .. pk respectively that divide n, i.e., we can write n as n = (p1a1)*(p2a2)* ... (pkak).Sum of divisors = (1 + p1 + p12 ... p1a1) * (1 + p2 + p22 ... p2a2) * ............................................. (1 + pk + pk2 ... pkak) We can notice that individual terms of above formula are Geometric Progressions (GP). Wecan rewrite the formula as.Sum of divisors = (p1a1+1 - 1)/(p1 -1) * (p2a2+1 - 1)/(p2 -1) * .................................. (pkak+1 - 1)/(pk -1)5. For a product of N numbers, if we have to subtract a constant K such that the product gets its maximum value, then subtract it from a largest value such that largest value-k is greater than 0. If we have to subtract a constant K such that the product gets its minimum value, then subtract it from the smallest value where smallest value-k should be greater than 06. Goldbach's conjecture: Every even integer greater than 2 can be expressed as the sum of 2 primes.7. Perfect numbers or Amicable numbers: Perfect numbers are those numbers which are equal to the sum of their proper divisors. Example: 6 = 1 + 2 + 38. Lychrel numbers: Are those numbers that cannot form a palindrome when repeatedly reversed and added to itself. For example 47 is not a Lychrel Number as 47 + 74 = 1219. Lemoine’s Conjecture : Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture.10. Fermat's Last Theorem : According to the theorem, no three positive integers a, b, c satisfy the equation, a^n + b^n = c^n for any integer value of n greater than 2. For n = 1 and n = 2, the equation have infinitely many solutions.Number Theory AlgorithmsGCD and LCM GCD and LCMLCM of arrayGCD of arrayBasic and Extended Euclidean algorithmsRecent Articles on GCD and LCM!Prime Factorization and Divisors :Prime factorsPollard’s Rho Algorithm for Prime FactorizationFind all divisors of a natural numberSum of all proper divisors of a natural numberPrime Factorization using Sieve O(log n) for multiple queriesFind politeness of a numberPrint prime numbers in a given range using C++ STLk-th prime factor of a given numberSmith NumbersRecent Articles on Prime Factors!Fibonacci Numbers:Fibonacci NumbersInteresting facts about Fibonacci numbersHow to check if a given number is Fibonacci number?Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation)Recent Articles on Fibonacci Numbers!Catalan Numbers :Catalan numbersApplications of Catalan NumbersRecent Articles on Catalan Numbers!Modular Arithmetic :Modular Exponentiation (Power in Modular Arithmetic)Modular multiplicative inverseModular DivisionMultiplicative orderFind Square Root under Modulo p | Set 1 (When p is in form of 4*i + 3)Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)Euler’s criterion (Check if square root under modulo p exists)Multiply large integers under large moduloFind sum of modulo K of first N natural numberHow to compute mod of a big number?BigInteger Class in JavaModulo 10^9+7 (1000000007)How to avoid overflow in modular multiplication?RSA Algorithm in CryptographyFind (a^b)%m where ‘a’ is very largeFind power of power under mod of a primeRecent Articles on Modular Arithmetic!Euler Totient Function:Euler’s Totient FunctionOptimized Euler Totient Function for Multiple EvaluationsEuler’s Totient function for all numbers smaller than or equal to nPrimitive root of a prime number n modulo nnCr Computations:Binomial CoefficientCompute nCr % p | Set 1 (Introduction and Dynamic Programming Solution)Compute nCr % p | Set 2 (Lucas Theorem)Compute nCr % p | Set 3 (Using Fermat Little Theorem)Chinese Remainder Theorem: Set 1 (Introduction)Set 2 (Inverse Modulo based Implementation)Cyclic Redundancy Check and Modulo-2 DivisionUsing Chinese Remainder Theorem to Combine Modular equationsFactorial:FactorialLegendre’s formula (Given p and n, find the largest x such that p^x divides n!)Sum of divisors of factorial of a numberCount Divisors of FactorialCompute n! under modulo pWilson’s TheoremPrimality Test | Set 1 (Introduction and School Method)Primality Test | Set 2 (Fermat Method)Primality Test | Set 3 (Miller–Rabin)Primality Test | Set 4 (Solovay-Strassen)GFact 22 | (2^x + 1 and Prime)Euclid's LemmaSieve of EratosthenesSegmented SieveSieve of AtkinSieve of Sundaram to print all primes smaller than nSieve of Eratosthenes in 0(n) time complexityCheck if a large number is divisible by 3 or notCheck if a large number is divisible by 11 or notTo check divisibility of any large number by 999Carmichael NumbersGenerators of finite cyclic group under additionMeasure one litre using two vessels and infinite water supplyProgram to find last digit of n’th Fibonacci NumberGCD of two numbers when one of them can be very largeFind Last Digit Of a^b for Large NumbersRemainder with 7 for large numbersPartition a number into two divisible partsNumber of substrings divisible by 6 in a string of integers'Practice Problems' on Modular ArithmeticPadovan, OESIS Comment More infoAdvertise with us Next Article Analysis of Algorithms K kartik Follow Improve Article Tags : Mathematical DSA number-theory binomial coefficient Practice Tags : Mathematicalnumber-theory Similar Reads Basics & PrerequisitesLogic Building ProblemsLogic building is about creating clear, step-by-step methods to solve problems using simple rules and principles. 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