Let's start with Divisors and Multiples and work through each subtopic. I'll provide:
- Theory: Brief explanation of the concept.
- Practical Application: Where and why the concept is used.
- Implementation in C++: Example code for clarity.
1. Divisors and Multiples
Theory:
- Factors: Numbers that divide a given number ( n ) without leaving a remainder.
- Example: Factors of 12 are {1, 2, 3, 4, 6, 12}.
- Greatest Common Divisor (GCD): The largest number that divides two numbers ( a ) and ( b ).
- Least Common Multiple (LCM): The smallest number divisible by both ( a ) and ( b ).
Practical Application:
- GCD is used in simplifying fractions.
- LCM is useful for scheduling problems, finding periods, etc.
- Factors are fundamental in number theory, especially in problems involving divisors.
Implementation in C++:
#include <iostream>
#include <vector>
using namespace std;
// Function to find factors of a number
vector<int> findFactors(int n) {
vector<int> factors;
for (int i = 1; i * i <= n; ++i) {
if (n % i == 0) {
factors.push_back(i); // Add the factor
if (i != n / i) { // Avoid duplicate when n is a perfect square
factors.push_back(n / i);
}
}
}
return factors;
}
// Function to compute GCD using Euclid's Algorithm
int gcd(int a, int b) {
while (b != 0) {
int temp = b;
b = a % b;
a = temp;
}
return a;
}
// Function to compute LCM using GCD
int lcm(int a, int b) {
return (a / gcd(a, b)) * b; // Formula: (a * b) / GCD
}
int main() {
int num = 12;
cout << "Factors of " << num << ": ";
vector<int> factors = findFactors(num);
for (int factor : factors) {
cout << factor << " ";
}
cout << endl;
int a = 24, b = 36;
cout << "GCD of " << a << " and " << b << ": " << gcd(a, b) << endl;
cout << "LCM of " << a << " and " << b << ": " << lcm(a, b) << endl;
return 0;
}
Output:
Factors of 12: 1 12 2 6 3 4
GCD of 24 and 36: 12
LCM of 24 and 36: 72
