/** * @file * @brief C++ Program to calculate the number of positive divisors * * This algorithm uses the prime factorization approach. * Any positive integer can be written as a product of its prime factors. *
Let \f$N = p_1^{e_1} \times p_2^{e_2} \times\cdots\times p_k^{e_k}\f$ * where \f$p_1,\, p_2,\, \dots,\, p_k\f$ are distinct prime factors of \f$N\f$ and * \f$e_1,\, e_2,\, \dots,\, e_k\f$ are respective positive integer exponents. *
Each positive divisor of \f$N\f$ is in the form * \f$p_1^{g_1}\times p_2^{g_2}\times\cdots\times p_k^{g_k}\f$ * where \f$0\le g_i\le e_i\f$ are integers for all \f$1\le i\le k\f$. *
Finally, there are \f$(e_1+1) \times (e_2+1)\times\cdots\times (e_k+1)\f$ * positive divisors of \f$N\f$ since we can choose every \f$g_i\f$ * independently. * * Example: *
\f$N = 36 = (3^2 \cdot 2^2)\f$ *
\f$\mbox{number_of_positive_divisors}(36) = (2+1) \cdot (2+1) = 9\f$. *
list of positive divisors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36. * * Similarly, for N = -36 the number of positive divisors remain same. **/ #include #include /** * Function to compute the number of positive divisors. * @param n number to compute divisors for * @returns number of positive divisors of n (or 1 if n = 0) */ int number_of_positive_divisors(int n) { if (n < 0) { n = -n; // take the absolute value of n } int number_of_divisors = 1; for (int i = 2; i * i <= n; i++) { // This part is doing the prime factorization. // Note that we cannot find a composite divisor of n unless we would // already previously find the corresponding prime divisor and dvided // n by that prime. Therefore, all the divisors found here will // actually be primes. // The loop terminates early when it is left with a number n which // does not have a divisor smaller or equal to sqrt(n) - that means // the remaining number is a prime itself. int prime_exponent = 0; while (n % i == 0) { // Repeatedly divide n by the prime divisor n to compute // the exponent (e_i in the algorithm description). prime_exponent++; n /= i; } number_of_divisors *= prime_exponent + 1; } if (n > 1) { // In case the remaining number n is a prime number itself // (essentially p_k^1) the final answer is also multiplied by (e_k+1). number_of_divisors *= 2; } return number_of_divisors; } /** * Test implementations */ void tests() { assert(number_of_positive_divisors(36) == 9); assert(number_of_positive_divisors(-36) == 9); assert(number_of_positive_divisors(1) == 1); assert(number_of_positive_divisors(2011) == 2); // 2011 is a prime assert(number_of_positive_divisors(756) == 24); // 756 = 2^2 * 3^3 * 7 } /** * Main function */ int main() { tests(); int n; std::cin >> n; if (n == 0) { std::cout << "All non-zero numbers are divisors of 0 !" << std::endl; } else { std::cout << "Number of positive divisors is : "; std::cout << number_of_positive_divisors(n) << std::endl; } return 0; }