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Largest factor of a given number which is a perfect square

  • Difficulty Level : Medium
  • Last Updated : 21 Sep, 2021

Given a number N      . The task is to find the largest factor of that number which is a perfect square.
Examples

Input : N = 420
Output : 4

Input : N = 100
Output : 100

 

A Simple Solution is to traverse all of the numbers in decreasing order from the given number down till 1 and if any of these numbers is a factor of the given number and is also a perfect square, print that number.
Time Complexity: O(N)
Better Solution : A better solution is to use prime factorization of the given number.

  • First find all the prime factors of that number till sqrt(num).
  • Take a variable, answer and initialize it to 1. It represents the largest square number which is also a factor of that number.
  • Now, Check If any prime number occurs even number of times in the prime factorization of the given number, if yes then multiply the answer with that prime factor the number of times it occurs.
  • Else, if it occurs odd number of times then multiply the answer with prime number (K – 1) times, K is the frequency of that prime factor in the prime factorization.

Below is the implementation of the above approach: 

C++




// C++ program to find the largest factor of
// a number which is also a perfect square
 
#include <cmath>
#include <iostream>
using namespace std;
 
// Function to find the largest factor
// of a given number which
// is a perfect square
int largestSquareFactor(int num)
{
    // Initialise the answer to 1
    int answer = 1;
 
    // Finding the prime factors till sqrt(num)
    for (int i = 2; i < sqrt(num); ++i) {
        // Frequency of the prime factor in the
        // factorisation initialised to 0
        int cnt = 0;
        int j = i;
 
        while (num % j == 0) {
            cnt++;
            j *= i;
        }
 
        // If the frequency is odd then multiply i
        // frequency-1 times to the answer
        if (cnt & 1) {
            cnt--;
            answer *= pow(i, cnt);
        }
        // Else if it is even, multiply
        // it frequency times
        else {
            answer *= pow(i, cnt);
        }
    }
 
    return answer;
}
 
// Driver Code
int main()
{
    int N = 420;
 
    cout << largestSquareFactor(N);
 
    return 0;
}

Java




// Java program to find the largest factor of
// a number which is also a perfect square
 
class GFG
{
   
// Function to find the largest factor
// of a given number which
// is a perfect square
static int largestSquareFactor(int num)
{
    // Initialise the answer to 1
    int answer = 1;
   
    // Finding the prime factors till sqrt(num)
    for (int i = 2; i < Math.sqrt(num); ++i) {
        // Frequency of the prime factor in the
        // factorisation initialised to 0
        int cnt = 0;
        int j = i;
   
        while (num % j == 0) {
            cnt++;
            j *= i;
        }
   
        // If the frequency is odd then multiply i
        // frequency-1 times to the answer
        if ((cnt & 1)!=0) {
            cnt--;
            answer *= Math.pow(i, cnt);
        }
        // Else if it is even, multiply
        // it frequency times
        else {
            answer *= Math.pow(i, cnt);
        }
    }
   
    return answer;
}
   
// Driver Code
public static void  main(String args[])
{
    int N = 420;
   
    System.out.println(largestSquareFactor(N));
   
}
}

Python 3




# Python 3 program to find the largest
# factor of a number which is also a
# perfect square
import math
 
# Function to find the largest factor
# of a given number which is a
# perfect square
def largestSquareFactor( num):
 
    # Initialise the answer to 1
    answer = 1
 
    # Finding the prime factors till sqrt(num)
    for i in range(2, int(math.sqrt(num))) :
         
        # Frequency of the prime factor in the
        # factorisation initialised to 0
        cnt = 0
        j = i
 
        while (num % j == 0) :
            cnt += 1
            j *= i
 
        # If the frequency is odd then multiply i
        # frequency-1 times to the answer
        if (cnt & 1) :
            cnt -= 1
            answer *= pow(i, cnt)
         
        # Else if it is even, multiply
        # it frequency times
        else :
            answer *= pow(i, cnt)
    return answer
 
# Driver Code
if __name__ == "__main__":
    N = 420
 
    print(largestSquareFactor(N))
 
# This code is contributed
# by ChitraNayal

C#




// C# program to find the largest factor of
// a number which is also a perfect square
using System;
 
class GFG
{
     
// Function to find the largest factor of
// a given number which is a perfect square
static double largestSquareFactor(double num)
{
    // Initialise the answer to 1
    double answer = 1;
 
    // Finding the prime factors
    // till sqrt(num)
    for (int i = 2; i < Math.Sqrt(num); ++i)
    {
        // Frequency of the prime factor in
        // the factorisation initialised to 0
        int cnt = 0;
        int j = i;
 
        while (num % j == 0)
        {
            cnt++;
            j *= i;
        }
 
        // If the frequency is odd then multiply i
        // frequency-1 times to the answer
        if ((cnt & 1) != 0)
        {
            cnt--;
            answer *= Math.Pow(i, cnt);
        }
         
        // Else if it is even, multiply
        // it frequency times
        else
        {
            answer *= Math.Pow(i, cnt);
        }
    }
 
    return answer;
}
 
// Driver Code
static public void Main ()
{
    int N = 420;
    Console.WriteLine(largestSquareFactor(N));
}
}
 
// This code is contributed by Sach_Code

PHP




<?php
// PHP program to find the largest
// factor of a number which is also
// a perfect square
 
// Function to find the largest
// factor of a given number which
// is a perfect square
function largestSquareFactor($num)
{
    // Initialise the answer to 1
    $answer = 1;
 
    // Finding the prime factors
    // till sqrt(num)
    for ($i = 2; $i < sqrt($num); ++$i)
    {
        // Frequency of the prime factor
        // in the factorisation initialised to 0
        $cnt = 0;
        $j = $i;
 
        while ($num % $j == 0)
        {
            $cnt++;
            $j *= $i;
        }
 
        // If the frequency is odd then
        // multiply i frequency-1 times
        // to the answer
        if ($cnt & 1)
        {
            $cnt--;
            $answer *= pow($i, $cnt);
        }
         
        // Else if it is even, multiply
        // it frequency times
        else
        {
            $answer *= pow($i, $cnt);
        }
    }
 
    return $answer;
}
 
// Driver Code
$N = 420;
echo largestSquareFactor($N);
 
// This code is contributed
// by Sach_Code
?>

Javascript




<script>
// Javascript program to find the largest factor of
// a number which is also a perfect square
 
// Function to find the largest factor
// of a given number which
// is a perfect square
function largestSquareFactor(num)
{
 
    // Initialise the answer to 1
    let answer = 1;
 
    // Finding the prime factors till sqrt(num)
    for (let i = 2; i < Math.sqrt(num); ++i)
    {
     
        // Frequency of the prime factor in the
        // factorisation initialised to 0
        let cnt = 0;
        let j = i;
 
        while (num % j == 0) {
            cnt++;
            j *= i;
        }
 
        // If the frequency is odd then multiply i
        // frequency-1 times to the answer
        if (cnt & 1)
        {
            cnt--;
            answer *= Math.pow(i, cnt);
        }
         
        // Else if it is even, multiply
        // it frequency times
        else {
            answer *= Math.pow(i, cnt);
        }
    }
 
    return answer;
}
 
// Driver Code
let N = 420;
 
document.write(largestSquareFactor(N));
 
// This code is contributed by souravmahato348.
</script>
Output: 
4

 

Time Complexity: O( sqrt(N) )




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