Prime Factors

Definition

Definition

The Prime Factors Algorithm is a computational method used to find the prime factors of a given integer. It leverages the fundamental theorem of arithmetic, which states that every integer greater than 1 either is a prime number itself or can be factorized into a unique combination of prime numbers. The algorithm efficiently identifies these prime factors, aiding in various mathematical and cryptographic applications

Explanation

Iteratively dividing the given integer by the smallest prime numbers (starting from 2) until the quotient becomes 1. At each step, it checks if the current divisor divides the integer evenly. If so, the divisor is a prime factor, and it is added to the list of factors. This process continues until the quotient becomes 1, indicating that all prime factors have been found

Guidance

• Start with the given integer to factorize
• Initialize an empty list to store prime factors
• Begin iterating from 2, the smallest prime number
  • check if the current number divides the given integer evenly
  • if it does, add the current number to the list of prime factors
  • divide the given integer by the current number
  • repeat steps until the given integer becomes 1
• Return the list of prime factors

Tips

• use the fundamental theorem of arithmetic to guide the algorithm
• utilize efficient data structures to store factors and perform divisions
• handle edge cases like prime numbers or negative integers appropriately

Practice

Practice

prime_factors(n):
  factors = []  # Initialize empty list to store prime factors
  divisor = 2   # Start with the smallest prime number

  while n > 1:
    while n % divisor == 0:  # Check if divisor divides n evenly
      factors.append(divisor)  # Add divisor to list of prime factors
      n = n / divisor   # Update n by dividing by divisor
    divisor = divisor + 1  # Move to the next number
  return factors  # Return the list of prime factors

Solution

package main

import (
	"math"
)

func primeFactors(n int) []int {
	var factors []int

	for n%2 == 0 {
		factors = append(factors, 2)
		n /= 2
	}

	for factor := 3; factor <= int(math.Sqrt(float64(n))); factor += 2 {
		for n%factor == 0 {
			factors = append(factors, factor)
			n /= factor
		}
	}

	if n > 2 {
		factors = append(factors, n)
	}

	return factors
}

import java.util.ArrayList;
import java.util.List;

public class Main {

  public static List<Integer> primeFactors(int n) {
    List<Integer> factors = new ArrayList<>();

    while (n % 2 == 0) {
      factors.add(2);
      n /= 2;
    }

    for (int factor = 3; factor <= Math.sqrt(n); factor += 2) {
      while (n % factor == 0) {
        factors.add(factor);
        n /= factor;
      }
    }

    if (n > 2) {
      factors.add(n);
    }

    return factors;
  }
}

function primeFactors(n) {
  const factors = [];

  while (n % 2 === 0) {
    factors.push(2);
    n = Math.floor(n / 2);
  }

  for (let factor = 3; factor <= Math.sqrt(n); factor += 2) {
    while (n % factor === 0) {
      factors.push(factor);
      n = Math.floor(n / factor);
    }
  }

  if (n > 2) {
    factors.push(n);
  }

  return factors;
}

fun primeFactors(n: Int): List<Int> {
    val factors = mutableListOf<Int>()

    var num = n
    while (num % 2 == 0) {
        factors.add(2)
        num /= 2
    }

    var factor = 3
    while (factor <= Math.sqrt(num.toDouble())) {
        while (num % factor == 0) {
            factors.add(factor)
            num /= factor
        }
        factor += 2
    }

    if (num > 2) {
        factors.add(num)
    }

    return factors
}

def prime_factors(n):
    factors = []

    while n % 2 == 0:
        factors.append(2)
        n //= 2

    factor = 3
    while factor <= int(n ** 0.5):
        while n % factor == 0:
            factors.append(factor)
            n //= factor
        factor += 2

    if n > 2:
        factors.append(n)

    return factors

fn prime_factors(mut n: i32) -> Vec<i32> {
    let mut factors = Vec::new();

    while n % 2 == 0 {
        factors.push(2);
        n /= 2;
    }

    let mut factor = 3;
    while factor <= (n as f64).sqrt() as i32 {
        while n % factor == 0 {
            factors.push(factor);
            n /= factor;
        }
        factor += 2;
    }

    if n > 2 {
        factors.push(n);
    }

    factors
}

function primeFactors(n: number): number[] {
  const factors: number[] = [];

  while (n % 2 === 0) {
    factors.push(2);
    n = Math.floor(n / 2);
  }

  for (let factor = 3; factor <= Math.sqrt(n); factor += 2) {
    while (n % factor === 0) {
      factors.push(factor);
      n = Math.floor(n / factor);
    }
  }

  if (n > 2) {
    factors.push(n);
  }

  return factors;
}