Step by Step solution about How to resolve the algorithm Product of min and max prime factors step by step in the C++ programming language

The C++ program determines the prime factorization of numbers from 1 to 100 and calculates the product of the smallest and largest prime factors for each number. Here's a detailed explanation of the code:

1. Header Inclusions:

   • The program includes the necessary headers:
     • <iomanip>: Used for manipulating input and output.
     • <iostream>: For input/output operations.
     • <utility>: For using the std::pair data structure.

2. Function min_max_prime_factors:

   • This function takes an unsigned integer n as input and returns a pair of unsigned integers containing the minimum and maximum prime factors of n.
   • It initializes min_factor and max_factor to 1.
   • It checks if n is even (divisible by 2) using (n & 1) == 0. If so, it repeatedly divides n by 2 while it's even, and updates min_factor and max_factor to 2.
   • It then iterates through prime numbers starting from 3, incrementing by 2 in each iteration (p += 2).
   • For each prime number p, it checks if n is divisible by p. If so, it repeatedly divides n by p while it's divisible by p, updating min_factor to p if it's the first prime factor found, and updating max_factor to p.
   • If n is still greater than 1 after the loop, it means that n itself is a prime factor. In this case, it updates min_factor and max_factor to n if they're still 1.
   • Finally, the function returns a std::pair containing min_factor and max_factor.

3. Main Function:

   • The main function calculates the product of the smallest and greatest prime factors for numbers from 1 to 100.

   • It uses a for loop to iterate through numbers from 1 to 100.

   • For each number n, it calls min_max_prime_factors(n) to get the pair of prime factors.

   • It then calculates the product of the prime factors and prints it, formatted with a width of 4.

   • It prints a newline every 10 numbers to improve readability.

4. Output:

   • The program prints the product of the smallest and greatest prime factors of numbers from 1 to 100, each on a separate line. For example, for n=6, the smallest prime factor is 2 and the greatest prime factor is 3, so the product is 2 * 3 = 6. For n=12, the smallest prime factor is 2 and the greatest is 3, so the product is 2 * 3 = 6, and so on.

#include <iomanip>
#include <iostream>
#include <utility>

auto min_max_prime_factors(unsigned int n) {
    unsigned int min_factor = 1;
    unsigned int max_factor = 1;
    if ((n & 1) == 0) {
        while ((n & 1) == 0)
            n >>= 1;
        min_factor = 2;
        max_factor = 2;
    }
    for (unsigned int p = 3; p * p <= n; p += 2) {
        if (n % p == 0) {
            while (n % p == 0)
                n /= p;
            if (min_factor == 1)
                min_factor = p;
            max_factor = p;
        }
    }
    if (n > 1) {
        if (min_factor == 1)
            min_factor = n;
        max_factor = n;
    }
    return std::make_pair(min_factor, max_factor);
}

int main() {
    std::cout << "Product of smallest and greatest prime factors of n for 1 to "
                 "100:\n";
    for (unsigned int n = 1; n <= 100; ++n) {
        auto p = min_max_prime_factors(n);
        std::cout << std::setw(4) << p.first * p.second
                  << (n % 10 == 0 ? '\n' : ' ');
    }
}