Check if a number can be expressed as product of a prime and a composite number
Last Updated :
22 Aug, 2023
Given a number N, the task is to check if N can be represented as the product of a prime and a composite number or not. If it can, then print Yes, otherwise No.
Examples:
Input: N = 52
Output: Yes
Explanation: 52 can be represented as the multiplication of 4 and 13, where 4 is a composite and 13 is a prime number.
Input: N = 49
Output: No
Approach: This problem can be solved with the help of the Sieve of Eratosthenes algorithm. Now, to solve this problem, follow the below steps:
- Create a boolean array isPrime, where the ith element is true if it is a prime, otherwise it's false.
- Find all prime numbers till N using sieve algorithm.
- Now run a loop for i=2 to i<N, and on each iteration:
- Check for these two conditions:
- If N is divisible by i.
- If i is a prime number and N/i isn't or if i isn't a prime number and N/i is.
- If both of the above conditions satisfy, return true.
- Otherwise, return false.
- Print the answer, according to the above observation.
Below is the implementation of the above approach.
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to generate all prime
// numbers less than N
void SieveOfEratosthenes(int N, bool isPrime[])
{
// Initialize all entries of boolean array
// as true. A value in isPrime[i] will finally
// be false if i is Not a prime, else true
// bool isPrime[N+1];
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= N; i++)
isPrime[i] = true;
for (int p = 2; p * p <= N; p++) {
// If isPrime[p] is not changed,
// then it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= N; i += p)
isPrime[i] = false;
}
}
}
// Function to check if we can
// represent N as product of a prime
// and a composite number or not
bool isRepresentable(int N)
{
// Generating primes using Sieve
bool isPrime[N + 1];
SieveOfEratosthenes(N, isPrime);
// Traversing through the array
for (int i = 2; i < N; i++) {
if (N % i == 0) {
if (N % i == 0
and (isPrime[i] and !isPrime[N / i])
or (!isPrime[i] and isPrime[N / i])) {
return true;
}
}
}
return false;
}
// Driver Code
int main()
{
int N = 52;
if (isRepresentable(N)) {
cout << "Yes";
}
else {
cout << "No";
}
return 0;
}
// Java program to implement the above approach
import java.util.*;
public class GFG
{
// Function to generate all prime
// numbers less than N
static void SieveOfEratosthenes(int N, boolean []isPrime)
{
// Initialize all entries of boolean array
// as true. A value in isPrime[i] will finally
// be false if i is Not a prime, else true
// bool isPrime[N+1];
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= N; i++)
isPrime[i] = true;
for (int p = 2; p * p <= N; p++) {
// If isPrime[p] is not changed,
// then it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= N; i += p)
isPrime[i] = false;
}
}
}
// Function to check if we can
// represent N as product of a prime
// and a composite number or not
static boolean isRepresentable(int N)
{
// Generating primes using Sieve
boolean []isPrime = new boolean[N + 1];
SieveOfEratosthenes(N, isPrime);
// Traversing through the array
for (int i = 2; i < N; i++) {
if (N % i == 0) {
if (N % i == 0
&& (isPrime[i] && !isPrime[N / i])
|| (!isPrime[i] && isPrime[N / i])) {
return true;
}
}
}
return false;
}
// Driver Code
public static void main(String arg[])
{
int N = 52;
if (isRepresentable(N)) {
System.out.println("Yes");
}
else {
System.out.println("No");
}
}
}
// This code is contributed by Samim Hossain Mondal.
# python program for the above approach
import math
# Function to generate all prime
# numbers less than N
def SieveOfEratosthenes(N, isPrime):
# Initialize all entries of boolean array
# as true. A value in isPrime[i] will finally
# be false if i is Not a prime, else true
# bool isPrime[N+1];
isPrime[0] = False
isPrime[1] = False
for i in range(2, N+1):
isPrime[i] = True
for p in range(2, int(math.sqrt(N)) + 1):
# If isPrime[p] is not changed,
# then it is a prime
if (isPrime[p] == True):
# Update all multiples of p
for i in range(p+2, N+1, p):
isPrime[i] = False
# Function to check if we can
# represent N as product of a prime
# and a composite number or not
def isRepresentable(N):
# Generating primes using Sieve
isPrime = [0 for _ in range(N + 1)]
SieveOfEratosthenes(N, isPrime)
# Traversing through the array
for i in range(2, N):
if (N % i == 0):
if (N % i == 0 and (isPrime[i] and not isPrime[N // i]) or (not isPrime[i] and isPrime[N // i])):
return True
return False
# Driver Code
if __name__ == "__main__":
N = 52
if (isRepresentable(N)):
print("Yes")
else:
print("No")
# This code is contributed by rakeshsahni
// C# program to implement the above approach
using System;
class GFG
{
// Function to generate all prime
// numbers less than N
static void SieveOfEratosthenes(int N, bool []isPrime)
{
// Initialize all entries of boolean array
// as true. A value in isPrime[i] will finally
// be false if i is Not a prime, else true
// bool isPrime[N+1];
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= N; i++)
isPrime[i] = true;
for (int p = 2; p * p <= N; p++) {
// If isPrime[p] is not changed,
// then it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= N; i += p)
isPrime[i] = false;
}
}
}
// Function to check if we can
// represent N as product of a prime
// and a composite number or not
static bool isRepresentable(int N)
{
// Generating primes using Sieve
bool []isPrime = new bool[N + 1];
SieveOfEratosthenes(N, isPrime);
// Traversing through the array
for (int i = 2; i < N; i++) {
if (N % i == 0) {
if (N % i == 0
&& (isPrime[i] && !isPrime[N / i])
|| (!isPrime[i] && isPrime[N / i])) {
return true;
}
}
}
return false;
}
// Driver Code
public static void Main()
{
int N = 52;
if (isRepresentable(N)) {
Console.Write("Yes");
}
else {
Console.Write("No");
}
}
}
// This code is contributed by Samim Hossain Mondal.
<script>
// Javascript program for the above approach
// Function to generate all prime
// numbers less than N
function SieveOfEratosthenes(N, isPrime)
{
// Initialize all entries of boolean array
// as true. A value in isPrime[i] will finally
// be false if i is Not a prime, else true
// bool isPrime[N+1];
isPrime[0] = isPrime[1] = false;
for (let i = 2; i <= N; i++)
isPrime[i] = true;
for (let p = 2; p * p <= N; p++) {
// If isPrime[p] is not changed,
// then it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (let i = p * 2; i <= N; i += p)
isPrime[i] = false;
}
}
}
// Function to check if we can
// represent N as product of a prime
// and a composite number or not
function isRepresentable(N)
{
// Generating primes using Sieve
let isPrime = [];
SieveOfEratosthenes(N, isPrime);
// Traversing through the array
for (let i = 2; i < N; i++) {
if (N % i == 0) {
if (N % i == 0
&& (isPrime[i] && !isPrime[N / i])
|| (!isPrime[i] && isPrime[N / i])) {
return true;
}
}
}
return false;
}
// Driver Code
let N = 52;
if (isRepresentable(N)) {
document.write("Yes");
}
else {
document.write("No");
}
// This code is contributed by Samim Hossain Mondal.
</script>
Time complexity: O(N*log(logN))
Auxiliary Space: O(N)
Another Approach
Below is the implementation of the above approach:
C++
#include <iostream>
#include <cmath>
// Function to check if a number is prime
bool isPrime(int n) {
if (n <= 1)
return false;
for (int i = 2; i <= sqrt(n); i++) {
if (n % i == 0)
return false;
}
return true;
}
// Function to check if N can be represented as the product of a prime and a composite number
bool isProductOfPrimeAndComposite(int N) {
if (N <= 3)
return false;
for (int i = 2; i <= sqrt(N); i++) {
if (N % i == 0) {
int factor1 = i;
int factor2 = N / i;
// Check if both factors are prime and composite numbers
if (isPrime(factor1) && !isPrime(factor2))
return true;
if (!isPrime(factor1) && isPrime(factor2))
return true;
}
}
return false;
}
// Driver Code
int main() {
int N=52;
if (isProductOfPrimeAndComposite(N))
std::cout << "Yes" << std::endl;
else
std::cout << "No" << std::endl;
return 0;
}
import java.util.*;
public class GFG {
// Function to check if a number is prime
static boolean isPrime(int n) {
if (n <= 1)
return false;
for (int i = 2; i <= Math.sqrt(n); i++) {
if (n % i == 0)
return false;
}
return true;
}
// Function to check if N can be represented as the product
// of a prime and a composite number
static boolean isProductOfPrimeAndComposite(int N) {
if (N <= 3)
return false;
for (int i = 2; i <= Math.sqrt(N); i++) {
if (N % i == 0) {
int factor1 = i;
int factor2 = N / i;
// Check if both factors are prime and composite numbers
if (isPrime(factor1) && !isPrime(factor2))
return true;
if (!isPrime(factor1) && isPrime(factor2))
return true;
}
}
return false;
}
// Driver Code
public static void main(String[] args) {
int N = 52;
if (isProductOfPrimeAndComposite(N))
System.out.println("Yes");
else
System.out.println("No");
// This Code Is Contributed By Shubham Tiwari.
}
}
import math
# Function to check if a number is prime
def is_prime(n):
if n <= 1:
return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
# Function to check if N can be represented as the
# product of a prime and a composite number
def is_product_of_prime_and_composite(N):
if N <= 3:
return False
for i in range(2, int(math.sqrt(N)) + 1):
if N % i == 0:
factor1 = i
factor2 = N // i
# Check if both factors are prime and composite numbers
if is_prime(factor1) and not is_prime(factor2):
return True
if not is_prime(factor1) and is_prime(factor2):
return True
return False
# Driver Code
if __name__ == "__main__":
N = 52
if is_product_of_prime_and_composite(N):
print("Yes")
else:
print("No")
# This Code Is Contributed By Shubham Tiwari.
using System;
class GFG
{
// Function to check if a number is prime
static bool IsPrime(int n)
{
if (n <= 1)
return false;
for (int i = 2; i <= Math.Sqrt(n); i++)
{
if (n % i == 0)
return false;
}
return true;
}
// Function to check if N can be represented as the
// product of a prime and a composite number
static bool IsProductOfPrimeAndComposite(int N)
{
if (N <= 3)
return false;
for (int i = 2; i <= Math.Sqrt(N); i++)
{
if (N % i == 0)
{
int factor1 = i;
int factor2 = N / i;
// Check if both factors are prime and
// composite numbers
if (IsPrime(factor1) && !IsPrime(factor2))
return true;
if (!IsPrime(factor1) && IsPrime(factor2))
return true;
}
}
return false;
}
static void Main(string[] args)
{
int N = 52;
if (IsProductOfPrimeAndComposite(N))
Console.WriteLine("Yes");
else
Console.WriteLine("No");
// This Code Is Contributed By Shubham Tiwari.
}
}
// Function to check if a number is prime
function isPrime(n) {
// If the number is less than or equal to 1, it is not prime
if (n <= 1)
return false;
// Check divisibility from 2 up to the square root of the number
for (let i = 2; i <= Math.sqrt(n); i++) {
if (n % i === 0)
// If the number is divisible by any number in this range, it is not prime
return false;
}
// If the number is not divisible by any number in the range, it is prime
return true;
}
// Function to check if N can be represented as the product of a prime and a composite number
function isProductOfPrimeAndComposite(N) {
// If N is less than or equal to 3, it cannot be represented as the product of a prime and a composite number
if (N <= 3)
return false;
// Check for factors from 2 up to the square root of N
for (let i = 2; i <= Math.sqrt(N); i++) {
if (N % i === 0) {
let factor1 = i;
let factor2 = N / i;
// If one factor is prime and the other is composite, return true
if (isPrime(factor1) && !isPrime(factor2))
return true;
if (!isPrime(factor1) && isPrime(factor2))
return true;
}
}
// If no such factors are found, return false
return false;
}
// Test the function with an example
let N = 52;
if (isProductOfPrimeAndComposite(N))
console.log("Yes");
else
console.log("No");
// This Code Is Contributed By Shubham Tiwari.
Time Complexity: O(sqrt(N))
Auxiliary Space: O(1)
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