Understanding Prime Numbers in C++

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A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. The smallest prime numbers are 2, 3, 5, 7, 11, and so on. Prime numbers are important in mathematics and computer science because they are used in various algorithms and encryption techniques.

Determining if a Number is Prime

#include <iostream>
#include <math.h>
using namespace std;
bool isPrime(int n) {
    if (n <= 1)
        return false;
    for (int i = 2; i <= sqrt(n); i++) {
        if (n % i == 0)
            return false;
    }
    return true;
}
int main() {
    int num;
    cout << "Enter a number: ";
    cin >> num;
    if (isPrime(num))
        cout << num << " is a prime number." << endl;
    else
        cout << num << " is not a prime number." << endl;
    system("pause");
    return 0;
}

Counting Prime Numbers in a Specified Range

#include <iostream>
#include <math.h>
using namespace std;
bool isPrime(int n) {
    if (n <= 1)
        return false;
    for (int i = 2; i <= sqrt(n); i++) {
        if (n % i == 0)
            return false;
    }
    return true;
}
int countPrimes(int start, int end) {
    int count = 0;
    for (int i = start; i <= end; i++) {
        if (isPrime(i))
            count++;
    }
    return count;
}
int main() {
    int start, end;
    cout << "Enter the starting number: ";
    cin >> start;
    cout << "Enter the ending number: ";
    cin >> end;
    int count = countPrimes(start, end);
    cout << "The number of prime numbers between " << start << " and " << end << " is: " << count << endl;
    system("pause");
    return 0;
}

Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. For example, 6 is a perfect number because its proper positive divisors are 1, 2, and 3, and their sum is equal to 6.

Determining if a Number is Perfect:

#include <iostream>
using namespace std;
int getDivisorsSum(int n) {
    int sum = 0;
    for (int i = 1; i <= n / 2; i++) {
        if (n % i == 0) {
            sum += i;
        }
    }
    return sum;
}
bool isPerfect(int n) {
    return n == getDivisorsSum(n);
}
int main() {
    int n;
    cout << "Enter a positive integer: ";
    cin >> n;
    if (isPerfect(n)) {
        cout << n << " is a perfect number." << endl;
    } else {
        cout << n << " is not a perfect number." << endl;
    }
    system("pause");
    return 0;
}

Amicable Numbers

An amicable number is a pair of two different numbers so related that the sum of the proper divisors of each is equal to the other number. For example, the smallest pair of amicable numbers is (220, 284).

Determining if a Pair of Numbers are Amicable:

#include <iostream>
#include <math.h>
using namespace std;
int getDivisorsSum(int n) {
    int sum = 1;
    for (int i = 2; i <= sqrt(n); i++) {
        if (n % i == 0) {
            if (i == (n / i))
                sum = sum + i;
            else
                sum = sum + (i + n / i);
        }
    }
    return sum;
}
bool isAmicable(int num1, int num2) {
    return (getDivisorsSum(num1) == num2 && getDivisorsSum(num2) == num1);
}
int main() {
    int num1, num2;
    cout << "Enter two different positive integers: ";
    cin >> num1 >> num2;
    if (isAmicable(num1, num2)) {
        cout << num1 << " and " << num2 << " are amicable numbers." << endl;
    } else {
        cout << num1 << " and " << num2 << " are not amicable numbers." << endl;
    }
    system("pause");
    return 0;
}

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