Following is a program to get the result of Euler’s Totient function for all numbers smaller than or equal to n when n is given.Programimport java.util.Scanner; public class EulerTotient {    public static int gcd(int a,int b){       int i, hcf = 0;          for(i = 1; i

According to Euler’s criterion a square root of n under modulo p exists if and only if a number num exists such that num%p is equal to n%p.Programimport java.util.Scanner; public class EulersCriterion {    public static void main(String args[]) {       Scanner sc = new Scanner(System.in);       System.out.println("Enter n value :");             int n = sc.nextInt();       System.out.println("Enter p value :");       int p = sc.nextInt();       n = n % p;       int flag = 0;            for ... Read More

Following is a Java program to find the divisors of factorials of a number.Programimport java.util.Scanner; public class DivisorsOfFactorial {    public static long fact(int i) {       if(i

You can calculate the exponent of the largest power of a PrimeNumber that divides the factorial n! using Legendre's formula.Programimport java.util.Scanner; public class LegendresFormula {    static int Largestpower(int n, int p) {       int ans = 0;       while (n > 0) {          n /= p;          ans += n;       }       return ans;    }    public static void main (String[] args) {       Scanner sc = new Scanner(System.in);       System.out.println("Enter the n value :");     ... Read More

Binomial coefficient (c(n, r) or nCr) is calculated using the formula n!/r!*(n-r)!. Following is the Java program find out the binomial coefficient of given integers.Programimport java.util.Scanner; public class BinomialCoefficient {    public static long fact(int i) {       if(i

Following is a Java program which prints the multiplicative order of given numbers.import java.util.Scanner;Programpublic class MultiplicativeOrder {    public static int gcd(int num1, int num2) {       if (num2 != 0) {          return gcd(num2, num1 % num2);       } else {          return num1;       }    }    static int multiplicativeOrder(int num1, int num2) {       if (gcd(num1, num2) != 1) {          return -1;       }       int res = 1;       int p ... Read More

The java.math.BigInteger.modInverse(BigInteger m) returns a BigInteger whose value is (this-1 mod m). Using this method you can calculate Modular multiplicative inverse for a given number.ProgramLive Demoimport java.math.*; public class BigIntegerDemo {    public static void main(String[] args) {       // create 3 BigInteger objects       BigInteger bi1, bi2, bi3;             // create a BigInteger exponent       BigInteger exponent = new BigInteger("2");       bi1 = new BigInteger("7");       bi2 = new BigInteger("20");             // perform modPow operation on bi1 using bi2 and exp       bi3 = bi1.modPow(exponent, bi2);       String str = bi1 + "^" +exponent+ " mod " + bi2 + " is " +bi3;             // print bi3 value       System.out.println( str );    } }Output7^2 mod 20 is 9

The java.math.BigInteger.modPow(BigInteger exponent, BigInteger m) returns a BigInteger whose value is (thisexponent mod m). Unlike pow, this method permits negative exponents. You can calculate the modular Exponentiation using this method.ProgramLive Demoimport java.math.*; public class BigIntegerDemo {    public static void main(String[] args) {       // create 3 BigInteger objects       BigInteger bi1, bi2, bi3;             // create a BigInteger exponent       BigInteger exponent = new BigInteger("2");       bi1 = new BigInteger("7");       bi2 = new BigInteger("20");             // perform ... Read More

The nth Catalan number in terms of binomial coefficients is calculated by the formula(n + k )/k where k varies from 2 to n and n ≥ 0. i.e.Cn = (2n)!/((n+1)!n!)Programpublic class NthCatalanNumber {    public static long fact(int i) {       if(i

The nth Catalan number in terms of binomial coefficients is calculated by the formula(n + k )/k where k varies from 2 to n and n ≥ 0. i.e.Cn = (2n)!/((n+1)!n!)Programpublic class CatalanNumbers {    public static long fact(int i) {       if(i