Find Values of P and Q Satisfying the Equation N = P^2.Q

In this tutorial, we will explore how to determine values of P and Q satisfying the given equation.

A whole number higher than 1 whose only elements are 1 and itself is referred to as a prime number. A whole number that may be split evenly into another number is referred to as a factor. 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 are the first few prime numbers. Composite numbers are those that have more than two components. Neither prime nor composite, number one.

The goal is to determine P and Q that satisfy the expression N = P^2.Q, where P and Q are prime numbers, provided a number N(1 ≤ N ≤ 9×1018). P and Q must be prime numbers.

Instances:

Input: N = 2023
Output: P = 17, Q = 7

Input: N = 175
Output: P = 5, Q = 7

The execution of the aforementioned concept is shown below:

C++ Program:

#include <bits/stdc++.h>
using namespace std;

void solvePQ(int n)
{
	long long int p = 0, q = 0;

	for (long long int i = 2; (i * i * i) <= n; i++){

		if (n % i != 0)

			continue;
		if ((n / i) % i == 0) {

			p = i;
			q = n / (i * i);
		}

		else {
			q = i;
			p = (long long int)round(sqrt(n / i));
		}
	}
	cout << "P is " << p << "\n"
		<< "Q is " << q << endl;
}

int main()
{

	long long int N;
	cout << "Put a value of N:" <<endl;
	cin >> N;
	return 0;
}

Output 1

(If we input the value of N as 2023)

Put a value of N:
2023
P is 17
Q is 7

Output 2

(If we input the value of N as 175)

Put a value of N:
175
P is 5
Q is 7

Output 3

(If we input the value of N as 158)

Put a value of N:
158
P is 9
Q is 2

Output 4

(If we input the value of N as 254)

Put a value of N:
254
P is 11
Q is 2

We can also write the above approach in javascript.

Javascript Code:

Program 1:

function solvePQ(n)
{
let p = 0;
let q = 0;

for (let i = 2; i * i * i <= n; i++)
{
if (n % i !== 0) {
continue;
}

if ((n / i) % i === 0) {
p = i;
q = n / (i * i);
}
else {
q = i;
p = Math.round(Math.sqrt(n / i));
}
}
console.log("P is " + p);
console.log("Q is " + q);
}
solvePQ(2023);

Output

P is 17
Q is 7

Program 2:

function solvePQ(n)
{
let p = 0;
let q = 0;

for (let i = 2; i * i * i <= n; i++)
{
if (n % i !== 0) {
continue;
}

if ((n / i) % i === 0) {
p = i;
q = n / (i * i);
}
else {
q = i;
p = Math.round(Math.sqrt(n / i));
}
}
console.log("P is " + p);
console.log("Q is " + q);
}
solvePQ(175);

Output