Understanding how to find factors of a number is a fundamental concept in mathematics and computer science. Factors are the integers that can be multiplied together to produce a given number. In this guide, we’ll explore the factors of the number 75 and provide code examples in different programming languages to demonstrate how to compute them programmatically.

Understanding Factors

Factors of a number are integers that divide the number exactly without leaving a remainder. In other words, if a and b are factors of n, then:

a × b = n

For example, the factors of 6 are 1, 2, 3, and 6 because:

  • 1 × 6 = 6
  • 2 × 3 = 6

Factors are essential in various areas such as simplifying fractions, finding greatest common divisors (GCD), and in prime factorization.

Factors of 75

Let’s determine the factors of 75.

  1. Start with 1 and 75:
  • 1 × 75 = 75
  1. Check for other divisors:
  • 2: 75 ÷ 2 = 37.5 → Not an integer.
  • 3: 75 ÷ 3 = 25 → Integer.
  • 4: 75 ÷ 4 = 18.75 → Not an integer.
  • 5: 75 ÷ 5 = 15 → Integer.
  • 6: 75 ÷ 6 = 12.5 → Not an integer.
  • 7: 75 ÷ 7 ≈ 10.714 → Not an integer.
  • 15: 75 ÷ 15 = 5 → Integer.
  • 25: 75 ÷ 25 = 3 → Integer.
  • 75: Already listed.

Complete List of Factors of 75:

1, 3, 5, 15, 25, 75

Finding Factors Programmatically

Finding factors programmatically involves iterating through a range of numbers and checking if they divide the target number without a remainder. Below are examples in various programming languages demonstrating how to find the factors of 75.

1 Python

# Python program to find factors of 75

def find_factors(n):
    factors = []
    for i in range(1, n + 1):
        if n % i == 0:
            factors.append(i)
    return factors

number = 75
print(f"Factors of {number} are: {find_factors(number)}")

Output:

Factors of 75 are: [1, 3, 5, 15, 25, 75]

Explanation:

  • The function find_factors iterates from 1 to n (inclusive).
  • It checks if n is divisible by i using the modulus operator %.
  • If divisible, i is appended to the factors list.

2 Java

// Java program to find factors of 75

public class FactorsOf75 {
    public static void main(String[] args) {
        int number = 75;
        System.out.print("Factors of " + number + " are: ");
        for(int i = 1; i <= number; i++) {
            if(number % i == 0) {
                System.out.print(i + " ");
            }
        }
    }
}

Output:

Factors of 75 are: 1 3 5 15 25 75

Explanation:

  • A for loop iterates from 1 to number.
  • The if statement checks divisibility.
  • If i is a factor, it’s printed out.

3 C++

// C++ program to find factors of 75

#include <iostream>
#include <vector>

using namespace std;

int main() {
    int number = 75;
    vector<int> factors;

    for(int i = 1; i <= number; ++i){
        if(number % i == 0){
            factors.push_back(i);
        }
    }

    cout << "Factors of " << number << " are: ";
    for(auto factor : factors){
        cout << factor << " ";
    }
    return 0;
}

Output:

Factors of 75 are: 1 3 5 15 25 75

Explanation:

  • Uses a vector to store factors.
  • Iterates and checks for divisibility.
  • Prints the factors after collecting them.

4 JavaScript

// JavaScript program to find factors of 75

function findFactors(n) {
    let factors = [];
    for(let i = 1; i <= n; i++) {
        if(n % i === 0) {
            factors.push(i);
        }
    }
    return factors;
}

let number = 75;
console.log(`Factors of ${number} are: ${findFactors(number).join(', ')}`);

Output:

Factors of 75 are: 1, 3, 5, 15, 25, 75

Explanation:

  • Defines a function findFactors that returns an array of factors.
  • Uses console.log to display the factors, joined by commas.

5 C

// C# program to find factors of 75

using System;
using System.Collections.Generic;

class FactorsOf75 {
    static void Main() {
        int number = 75;
        List<int> factors = new List<int>();

        for(int i = 1; i <= number; i++) {
            if(number % i == 0) {
                factors.Add(i);
            }
        }

        Console.WriteLine($"Factors of {number} are: {string.Join(", ", factors)}");
    }
}

Output:

Factors of 75 are: 1, 3, 5, 15, 25, 75

Explanation:

  • Uses a List<int> to store factors.
  • Iterates and checks for factors.
  • Prints the list using string.Join for formatting.

6 Ruby

# Ruby program to find factors of 75

def find_factors(n)
  factors = []
  (1..n).each do |i|
    factors << i if n % i == 0
  end
  factors
end

number = 75
puts "Factors of #{number} are: #{find_factors(number).join(', ')}"

Output:

Factors of 75 are: 1, 3, 5, 15, 25, 75

Explanation:

  • Defines a method find_factors that returns an array of factors.
  • Uses range (1..n) to iterate.
  • Appends to factors array if divisible.

Conclusion

Finding the factors of a number like 75 is a straightforward task that can be accomplished manually or programmatically. Understanding how to implement this in various programming languages not only reinforces fundamental programming concepts but also enhances problem-solving skills. Whether you’re using Python, Java, C++, JavaScript, C#, or Ruby, the logic remains consistent:

  1. Iterate through a range of numbers from 1 to the target number.
  2. Check if the target number is divisible by the current number in the iteration.
  3. Collect and display the factors accordingly.

This knowledge is essential for more advanced topics such as prime factorization, encryption algorithms, and algorithm optimization.

Feel free to adapt the provided code snippets to suit different numbers or integrate them into larger applications as needed.