Euclid's lemma Last Updated : 19 Nov, 2016 Summarize Comments Improve Suggest changes Share Like Article Like Report We are given two numbers x and y. We know that a number p divides their product. Can we say for sure that p also divides one of them? The answer is no. For example, consider x = 15, y = 6 and p = 9. p divides the product 15*6, but doesn't divide any of them. What if p is prime? Euclid's lemma states that if a prime p divides the product of two numbers (x*y), it must divide at least one of those numbers. For example x = 15, y = 6 and p = 5. p divides the product 15*6, it also divides 15. The idea is simple, since p is prime, it cannot be factorized. So it must either be completely present in x or in y. Generalization of Euclid's lemma: If p divides x*y and p is relatively prime to x, then p must divide y. In the above example, 5 is relatively prime to 6, therefore it must divide 15. Reference: https://en.wikipedia.org/wiki/Euclid's_lemma Comment More infoAdvertise with us Next Article Euclidean algorithms (Basic and Extended) K kartik Follow Improve Article Tags : Mathematical DSA Prime Number Practice Tags : MathematicalPrime Number Similar Reads Number Theory for DSA & Competitive Programming What is Number Theory?Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. It explores the fundamental nature of numbers and their mathematical structures. Number theory has been studied for centuries and has deep connectio 3 min read Number Theory (Interesting Facts and Algorithms) 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 repe 5 min read How to prepare for ACM - ICPC? ACM ICPC(Association for Computing Machinery - International Collegiate Programming Contest) is a worldwide annual multi-tiered programming contest being organized for over thirteen years. The contest is sponsored by IBM. This article focuses on what all topics that are important for competitive pro 7 min read Basics of Number TheoryProgram to Find GCD or HCF of Two NumbersGiven two positive integers a and b, the task is to find the GCD of the two numbers.Note: The GCD (Greatest Common Divisor) or HCF (Highest Common Factor) of two numbers is the largest number that divides both of them. Examples:Input: a = 20, b = 28Output: 4Explanation: The factors of 20 are 1, 2, 4 12 min read Program to find LCM of two numbersGiven two positive integers a and b. Find the Least Common Multiple (LCM) of a and b.LCM of two numbers is the smallest number which can be divided by both numbers. Input : a = 10, b = 5Output : 10Explanation : 10 is the smallest number divisible by both 10 and 5Input : a = 5, b = 11Output : 55Expla 5 min read Factorial of a NumberGiven the number n (n >=0), find its factorial. Factorial of n is defined as 1 x 2 x ... x n. For n = 0, factorial is 1. We are going to discuss iterative and recursive programs in this post.Examples:Input: n = 5Output: 120Explanation: 5! = 5 * 4 * 3 * 2 * 1 = 120Input: n = 4Output: 24Explanation 7 min read Print all prime factors of a given numberGiven a number n, the task is to find all prime factors of n.Examples:Input: n = 24Output: 2 2 2 3Explanation: The prime factorization of 24 is 23×3.Input: n = 13195Output: 5 7 13 29Explanation: The prime factorization of 13195 is 5×7×13×29.Approach:Every composite number has at least one prime fact 6 min read Binomial CoefficientGiven an integer values n and k, the task is to find the value of Binomial Coefficient C(n, k).A binomial coefficient C(n, k) can be defined as the coefficient of x^k in the expansion of (1 + x)^n.A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be 15+ min read Program for nth Catalan NumberCatalan numbers are defined as a mathematical sequence that consists of positive integers, which can be used to find the number of possibilities of various combinations. The nth term in the sequence denoted Cn, is found in the following formula: \frac{(2n)!}{((n + 1)! n!)} The first few Catalan numb 13 min read Euclid's lemmaWe are given two numbers x and y. We know that a number p divides their product. Can we say for sure that p also divides one of them? The answer is no. For example, consider x = 15, y = 6 and p = 9. p divides the product 15*6, but doesn't divide any of them. What if p is prime? Euclid's lemma states 1 min read Euclidean algorithms (Basic and Extended)The Euclidean algorithm is a way to find the greatest common divisor of two positive integers. GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors.Examples:input: a = 12, b = 20Output: 4Explanatio 9 min read Modular ArithmeticModular ArithmeticModular arithmetic is a system of arithmetic for numbers where numbers "wrap around" after reaching a certain value, called the modulus. It mainly uses remainders to get the value after wrap around. It is often referred to as "clock arithmetic. As you can see, the time values wrap after reaching 12 10 min read Modular AdditionModular addition is a basic math concept used in computers and number systems. It is commonly used in areas like cryptography (data security), coding, and digital signal processing. In modular addition, you add two numbers normally, but if the result reaches a certain fixed number (called the modulu 4 min read Modular MultiplicationGiven three integers a, b, and M, where M is the modulus. Compute the result of the modular multiplication of a and b under modulo M.((a×b) mod M)Examples:Input: a = 5, b = 3, M = 11Output: 4Explanation: a × b = 5 × 3 = 15, 15 % 11 = 4, so the result is 4.Input: a = 12, b = 15, M = 7Output: 5Explana 6 min read Modular DivisionGiven three positive integers a, b, and M, the objective is to find (a/b) % M i.e., find the value of (a × b-1 ) % M, where b-1 is the modular inverse of b modulo M.Examples: Input: a = 10, b = 2, M = 13Output: 5Explanation: The modular inverse of 2 modulo 13 is 7, so (10 / 2) % 13 = (10 × 7) % 13 = 13 min read Euler's Totient FunctionGiven an integer n, find the value of Euler's Totient Function, denoted as Φ(n). The function Φ(n) represents the count of positive integers less than or equal to n that are relatively prime to n. Euler's Totient function Φ(n) for an input n is the count of numbers in {1, 2, 3, ..., n-1} that are re 10 min read Euler's Totient function for all numbers smaller than or equal to nEuler's Totient function ?(n) for an input n is the count of numbers in {1, 2, 3, ..., n} that are relatively prime to n, i.e., the numbers whose GCD (Greatest Common Divisor) with n is 1. For example, ?(4) = 2, ?(3) = 2 and ?(5) = 4. There are 2 numbers smaller or equal to 4 that are relatively pri 13 min read Modular Exponentiation (Power in Modular Arithmetic)Given three integers x, n, and M, compute (x^n) % M (remainder when x raised to the power n is divided by M).Examples : Input: x = 3, n = 2, M = 4Output: 1Explanation: 32 % 4 = 9 % 4 = 1.Input: x = 2, n = 6, M = 10Output: 4Explanation: 26 % 10 = 64 % 10 = 4.Table of Content[Naive Approach] Repeated 6 min read Program to find remainder without using modulo or % operatorGiven two numbers 'num' and 'divisor', find remainder when 'num' is divided by 'divisor'. The use of modulo or % operator is not allowed.Examples : Input: num = 100, divisor = 7 Output: 2 Input: num = 30, divisor = 9 Output: 3 Method 1 : C++ // C++ program to find remainder without using // modulo o 9 min read Modular multiplicative inverseGiven two integers A and M, find the modular multiplicative inverse of A under modulo M.The modular multiplicative inverse is an integer X such that:A X ≡ 1 (mod M) Note: The value of X should be in the range {1, 2, ... M-1}, i.e., in the range of integer modulo M. ( Note that X cannot be 0 as A*0 m 15+ min read Multiplicative orderIn number theory, given an integer A and a positive integer N with gcd( A , N) = 1, the multiplicative order of a modulo N is the smallest positive integer k with A^k( mod N ) = 1. ( 0 < K < N ) Examples : Input : A = 4 , N = 7 Output : 3 explanation : GCD(4, 7) = 1 A^k( mod N ) = 1 ( smallest 7 min read Compute nCr%p using Lucas TheoremGiven three numbers n, r and p, compute the value of nCr mod p. Examples: Input: n = 10, r = 2, p = 13 Output: 6 Explanation: 10C2 is 45 and 45 % 13 is 6. Input: n = 1000, r = 900, p = 13 Output: 8 We strongly recommend referring below post as a prerequisite of this.Compute nCr % p | Set 1 (Introduc 12 min read Compute nCr%p using Fermat Little TheoremGiven three numbers n, r and p, compute the value of nCr mod p. Here p is a prime number greater than n. Here nCr is Binomial Coefficient.Example: Input: n = 10, r = 2, p = 13 Output: 6 Explanation: 10C2 is 45 and 45 % 13 is 6. Input: n = 6, r = 2, p = 13 Output: 2Recommended PracticenCrTry It! We h 15+ min read Introduction to Chinese Remainder TheoremWe are given two arrays num[0..k-1] and rem[0..k-1]. In num[0..k-1], every pair is coprime (gcd for every pair is 1). We need to find minimum positive number x such that: x % num[0] = rem[0], x % num[1] = rem[1], .......................x % num[k-1] = rem[k-1] Basically, we are given k numbers which 7 min read Implementation of Chinese Remainder theorem (Inverse Modulo based implementation)We are given two arrays num[0..k-1] and rem[0..k-1]. In num[0..k-1], every pair is coprime (gcd for every pair is 1). We need to find minimum positive number x such that: x % num[0] = rem[0], x % num[1] = rem[1], ....................... x % num[k-1] = rem[k-1] Example: Input: num[] = {3, 4, 5}, rem[ 11 min read Find Square Root under Modulo p | Set 1 (When p is in form of 4*i + 3)Given a number 'n' and a prime 'p', find square root of n under modulo p if it exists. It may be given that p is in the form for 4*i + 3 (OR p % 4 = 3) where i is an integer. Examples of such primes are 7, 11, 19, 23, 31, ... etc,Examples: Input: n = 2, p = 7Output: 3 or 4Explanation: 3 and 4 both a 14 min read Find Square Root under Modulo p | Set 2 (Shanks Tonelli algorithm)Given a number ‘n’ and a prime ‘p’, find square root of n under modulo p if it exists. Examples: Input: n = 2, p = 113 Output: 62 62^2 = 3844 and 3844 % 113 = 2 Input: n = 2, p = 7 Output: 3 or 4 3 and 4 both are square roots of 2 under modulo 7 because (3*3) % 7 = 2 and (4*4) % 7 = 2 Input: n = 2, 15+ min read Modular DivisionGiven three positive integers a, b, and M, the objective is to find (a/b) % M i.e., find the value of (a × b-1 ) % M, where b-1 is the modular inverse of b modulo M.Examples: Input: a = 10, b = 2, M = 13Output: 5Explanation: The modular inverse of 2 modulo 13 is 7, so (10 / 2) % 13 = (10 × 7) % 13 = 13 min read Cyclic Redundancy Check and Modulo-2 DivisionCyclic Redundancy Check or CRC is a method of detecting accidental changes/errors in the communication channel. CRC uses Generator Polynomial which is available on both sender and receiver side. An example generator polynomial is of the form like x3 + x + 1. This generator polynomial represents key 15+ min read Primitive root of a prime number n modulo nGiven a prime number n, the task is to find its primitive root under modulo n. The primitive root of a prime number n is an integer r between[1, n-1] such that the values of r^x(mod n) where x is in the range[0, n-2] are different. Return -1 if n is a non-prime number. Examples: Input : 7 Output : S 15 min read Euler's criterion (Check if square root under modulo p exists)Given a number 'n' and a prime p, find if square root of n under modulo p exists or not. A number x is square root of n under modulo p if (x*x)%p = n%p. Examples : Input: n = 2, p = 5 Output: false There doesn't exist a number x such that (x*x)%5 is 2 Input: n = 2, p = 7 Output: true There exists a 11 min read Using Chinese Remainder Theorem to Combine Modular equationsGiven N modular equations: A ? x1mod(m1) . . A ? xnmod(mn) Find x in the equation A ? xmod(m1*m2*m3..*mn) where mi is prime, or a power of a prime, and i takes values from 1 to n. The input is given as two arrays, the first being an array containing values of each xi, and the second array containing 12 min read Multiply large integers under large moduloGiven an integer a, b, m. Find (a * b ) mod m, where a, b may be large and their direct multiplication may cause overflow. However, they are smaller than half of the maximum allowed long long int value. Examples: Input: a = 426, b = 964, m = 235Output: 119Explanation: (426 * 964) % 235 = 410664 % 23 7 min read Compute n! under modulo pGiven a large number n and a prime p, how to efficiently compute n! % p?Examples : Input: n = 5, p = 13 Output: 3 5! = 120 and 120 % 13 = 3 Input: n = 6, p = 11 Output: 5 6! = 720 and 720 % 11 = 5 A Naive Solution is to first compute n!, then compute n! % p. This solution works fine when the value o 15+ min read Wilson's TheoremWilson's Theorem is a fundamental result in number theory that provides a necessary and sufficient condition for determining whether a given number is prime. It states that a natural number p > 1 is a prime number if and only if:(p - 1)! ≡ −1 (mod p)This means that the factorial of p - 1 (the pro 2 min read Number TheoryIntroduction to Primality Test and School MethodGiven a positive integer, check if the number is prime or not. A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of the first few prime numbers are {2, 3, 5, ...}Examples : Input: n = 11Output: trueInput: n = 15Output: falseInput: n = 1Output: 10 min read Fermat Method of Primality TestGiven a number n, check if it is prime or not. We have introduced and discussed the School method for primality testing in Set 1.Introduction to Primality Test and School MethodIn this post, Fermat's method is discussed. This method is a probabilistic method and is based on Fermat's Little Theorem. 10 min read Primality Test | Set 3 (Miller–Rabin)Given a number n, check if it is prime or not. We have introduced and discussed School and Fermat methods for primality testing.Primality Test | Set 1 (Introduction and School Method) Primality Test | Set 2 (Fermat Method)In this post, the Miller-Rabin method is discussed. This method is a probabili 15+ min read Solovay-Strassen method of Primality TestWe have already been introduced to primality testing in the previous articles in this series. Introduction to Primality Test and School MethodFermat Method of Primality TestPrimality Test | Set 3 (Miller–Rabin)The Solovay–Strassen test is a probabilistic algorithm used to check if a number is prime 13 min read Legendre's formula - Largest power of a prime p in n!Given an integer n and a prime number p, the task is to find the largest x such that px (p raised to power x) divides n!.Examples: Input: n = 7, p = 3Output: x = 2Explanation: 32 divides 7! and 2 is the largest such power of 3.Input: n = 10, p = 3Output: x = 4Explanation: 34 divides 10! and 4 is the 6 min read Carmichael NumbersA number n is said to be a Carmichael number if it satisfies the following modular arithmetic condition: power(b, n-1) MOD n = 1, for all b ranging from 1 to n such that b and n are relatively prime, i.e, gcd(b, n) = 1 Given a positive integer n, find if it is a Carmichael number. These numbers have 8 min read Number Theory | Generators of finite cyclic group under additionGiven a number n, find all generators of cyclic additive group under modulo n. Generator of a set {0, 1, ... n-1} is an element x such that x is smaller than n, and using x (and addition operation), we can generate all elements of the set.Examples: Input : 10 Output : 1 3 7 9 The set to be generated 5 min read Sum of divisors of factorial of a numberGiven a number n, we need to calculate the sum of divisors of factorial of the number. Examples: Input : 4 Output : 60 Factorial of 4 is 24. Divisors of 24 are 1 2 3 4 6 8 12 24, sum of these is 60. Input : 6 Output : 2418 A Simple Solution is to first compute the factorial of the given number, then 14 min read GFact | 2x + 1(where x > 0) is prime if and only if x is a power of 2A number of the form 2x + 1 (where x > 0) is prime if and only if x is a power of 2, i.e., x = 2n. So overall number becomes 22n + 1. Such numbers are called Fermat Number (Numbers of form 22n + 1). The first few Fermat numbers are 3, 5, 17, 257, 65537, 4294967297, .... An important thing to note 1 min read Sieve of EratosthenesGiven a number n, find all prime numbers less than or equal to n.Examples:Input: n = 10Output: [2, 3, 5, 7]Explanation: The prime numbers up to 10 obtained by Sieve of Eratosthenes are [2, 3, 5, 7].Input: n = 35Output: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]Explanation: The prime numbers up to 35 o 5 min read Program for Goldbach’s Conjecture (Two Primes with given Sum)Goldbach's conjecture is one of the oldest and best-known unsolved problems in the number theory of mathematics. Every even integer greater than 2 can be expressed as the sum of two primes. Examples: Input : n = 44 Output : 3 + 41 (both are primes) Input : n = 56 Output : 3 + 53 (both are primes) Re 12 min read Pollard's Rho Algorithm for Prime FactorizationGiven a positive integer n, and that it is composite, find a divisor of it.Example:Input: n = 12;Output: 2 [OR 3 OR 4]Input: n = 187;Output: 11 [OR 17]Brute approach: Test all integers less than n until a divisor is found. Improvisation: Test all integers less than ?nA large enough number will still 14 min read Game TheoryMinimax Algorithm in Game Theory | Set 1 (Introduction)Minimax is a kind of backtracking algorithm that is used in decision making and game theory to find the optimal move for a player, assuming that your opponent also plays optimally. It is widely used in two player turn-based games such as Tic-Tac-Toe, Backgammon, Mancala, Chess, etc.In Minimax the tw 9 min read Combinatorial Game Theory | Set 2 (Game of Nim)We strongly recommend to refer below article as a prerequisite of this. Combinatorial Game Theory | Set 1 (Introduction) In this post, Game of Nim is discussed. The Game of Nim is described by the following rules- “ Given a number of piles in which each pile contains some numbers of stones/coins. In 15+ min read Combinatorial Game Theory | Set 4 (Sprague - Grundy Theorem)Prerequisites : Grundy Numbers/Numbers and MexWe have already seen in Set 2 (https://www.geeksforgeeks.org/dsa/combinatorial-game-theory-set-2-game-nim/), that we can find who wins in a game of Nim without actually playing the game.Suppose we change the classic Nim game a bit. This time each player 15 min read Practice ProblemsRabin-Karp Algorithm for Pattern SearchingGiven two strings text and pattern string, your task is to find all starting positions where the pattern appears as a substring within the text. The strings will only contain lowercase English alphabets.While reporting the results, use 1-based indexing (i.e., the first character of the text is at po 12 min read Measure one litre using two vessels and infinite water supplyThere are two vessels of capacities 'a' and 'b' respectively. We have infinite water supply. Give an efficient algorithm to make exactly 1 litre of water in one of the vessels. You can throw all the water from any vessel any point of time. Assume that 'a' and 'b' are Coprimes.Following are the steps 15 min read Program to find last digit of n'th Fibonacci NumberGiven a number 'n', write a function that prints the last digit of n'th ('n' can also be a large number) Fibonacci number. Examples : Input : n = 0 Output : 0 Input: n = 2 Output : 1 Input : n = 7 Output : 3 Recommended PracticeThe Nth FibonnaciTry It! Method 1 : (Naive Method) Simple approach is to 13 min read GCD of two numbers when one of them can be very largeGiven two numbers 'a' and 'b' such that (0 <= a <= 10^12 and b <= b < 10^250). Find the GCD of two given numbers.Examples : Input: a = 978 b = 89798763754892653453379597352537489494736 Output: 6 Input: a = 1221 b = 1234567891011121314151617181920212223242526272829 Output: 3 Solution : In 9 min read Find Last Digit of a^b for Large NumbersYou are given two integer numbers, the base a (number of digits d, such that 1 <= d <= 1000) and the index b (0 <= b <= 922*10^15). You have to find the last digit of a^b.Examples: Input : 3 10Output : 9Input : 6 2Output : 6Input : 150 53Output : 0 After taking few examples, we can notic 9 min read Remainder with 7 for large numbersGiven a large number as a string, find the remainder of number when divided by 7. Examples : Input : num = 1234 Output : 2 Input : num = 1232 Output : 0 Input : num = 12345 Output : 4Recommended PracticeRemainder with 7Try It! Simple Approach is to convert a string into number and perform the mod op 8 min read Find (a^b)%m where 'a' is very largeGiven three numbers a, b and m where 1<=b,m<=10^6 and 'a' may be very large and contains upto 10^6 digits. The task is to find (a^b)%m. Examples: Input : a = 3, b = 2, m = 4 Output : 1 Explanation : (3^2)%4 = 9%4 = 1 Input : a = 987584345091051645734583954832576, b = 3, m = 11 Output: 10Recomm 15+ min read Find sum of modulo K of first N natural numberGiven two integer N ans K, the task is to find sum of modulo K of first N natural numbers i.e 1%K + 2%K + ..... + N%K. Examples : Input : N = 10 and K = 2. Output : 5 Sum = 1%2 + 2%2 + 3%2 + 4%2 + 5%2 + 6%2 + 7%2 + 8%2 + 9%2 + 10%2 = 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0 = 5.Recommended PracticeReve 9 min read Count sub-arrays whose product is divisible by kGiven an integer K and an array arr[], the task is to count all the sub-arrays whose product is divisible by K.Examples: Input: arr[] = {6, 2, 8}, K = 4 Output: 4 Required sub-arrays are {6, 2}, {6, 2, 8}, {2, 8}and {8}.Input: arr[] = {9, 1, 14}, K = 6 Output: 1 Naive approach: Run nested loops and 15+ min read Partition a number into two divisible partsGiven a number (as string) and two integers a and b, divide the string in two non-empty parts such that the first part is divisible by a and the second part is divisible by b. If the string can not be divided into two non-empty parts, output "NO", else print "YES" with the two parts. Examples: Input 15+ min read Find power of power under mod of a primeGiven four numbers A, B, C and M, where M is prime number. Our task is to compute A raised to power (B raised to power C) modulo M. Example: Input : A = 2, B = 4, C = 3, M = 23Output : 643 = 64 so,2^64(mod 23) = 6 A Naive Approach is to calculate res = BC and then calculate Ares % M by modular expon 7 min read Rearrange an array in maximum minimum form in O(1) extra spaceGiven a sorted array of positive integers, rearrange the array alternately i.e first element should be the maximum value, second minimum value, third-second max, fourth-second min and so on. Examples:Input: arr[] = {1, 2, 3, 4, 5, 6, 7} Output: arr[] = {7, 1, 6, 2, 5, 3, 4}Explanation: First 7 is th 8 min read Subset with no pair sum divisible by KGiven an array of integer numbers, we need to find maximum size of a subset such that sum of each pair of this subset is not divisible by K. Examples : Input : arr[] = [3, 7, 2, 9, 1] K = 3 Output : 3 Maximum size subset whose each pair sum is not divisible by K is [3, 7, 1] because, 3+7 = 10, 3+1 = 7 min read Number of substrings divisible by 6 in a string of integersGiven a string consisting of integers 0 to 9. The task is to count the number of substrings which when convert into integer are divisible by 6. Substring does not contain leading zeroes. Examples: Input : s = "606". Output : 5 Substrings "6", "0", "6", "60", "606" are divisible by 6. Input : s = "48 9 min read Miscellaneous Practice ProblemsHow to compute mod of a big number?Given a big number 'num' represented as string and an integer x, find value of "num % a" or "num mod a". Output is expected as an integer. Examples : Input: num = "12316767678678", a = 10 Output: num (mod a) ? 8 The idea is to process all digits one by one and use the property that xy (mod a) ? ((x 4 min read BigInteger Class in JavaBigInteger class is used for the mathematical operation which involves very big integer calculations that are outside the limit of all available primitive data types. In this way, BigInteger class is very handy to use because of its large method library and it is also used a lot in competitive progr 6 min read Modulo 10^9+7 (1000000007)In most programming competitions, we are required to answer the result in 10^9+7 modulo. The reason behind this is, if problem constraints are large integers, only efficient algorithms can solve them in an allowed limited time.What is modulo operation: The remainder obtained after the division opera 10 min read How to avoid overflow in modular multiplication?Consider below simple method to multiply two numbers. C++ #include <iostream> using namespace std; #define ll long long // Function to multiply two numbers modulo mod ll multiply(ll a, ll b, ll mod) { // Multiply two numbers and then take modulo to prevent // overflow return ((a % mod) * (b % 6 min read RSA Algorithm in CryptographyRSA(Rivest-Shamir-Adleman) Algorithm is an asymmetric or public-key cryptography algorithm which means it works on two different keys: Public Key and Private Key. The Public Key is used for encryption and is known to everyone, while the Private Key is used for decryption and must be kept secret by t 13 min read Like