Queries to find whether a number has exactly four distinct factors or not

Last Updated : 23 Jun, 2022

Given a positive integers ‘q’ and ‘n’. For each query ‘q’ find whether a number ‘n’ have exactly four distinct divisors or not. If the number have exactly four divisors then print ‘Yes’ else ‘No’.1 <= q, n <= 106

```Input:
2
10
12
Output:
Yes
No

Explanation:
For 1st query, n = 10 has exactly four divisor i.e., 1, 2, 5, 10.
For 2nd query, n = 12 has exactly six divisor i.e., 1, 2, 3, 4, 6, 12.```

Simple approach is to count factors by generating all divisors of a number by using this approach, after that check whether the count of all factors are equal to ‘4’ or not. Time complexity of this approach is O(sqrt(n)).
Better approach is to use Number theory. For a number to be have four factors, it must satisfy the following conditions:-

1. If number is a product of exactly two prime numbers(say p, q). Thus we can assure that it will have four factors i.e, 1, p, q, n.
2. If a number is cube of a prime number (or cube root of the number is prime). For example, let’s say n = 8, cube root = 2 that means ‘8’ can be written as 2*2*2 hence four factors are:- 1, 2, 4 and 8.

We can use sieve of Eratosthenes such that we will pre-calculate all the prime factor from 1 to 106. Now we will mark all numbers which are the product of two prime number by using two ‘for loops’ i.e, mark[p * q] =true. Meanwhile we will also mark all numbers(cube root) by taking cube of number i.e, mark[p * p * p] = true.
After that we can easily answer each query in O(1) time.
Below is pseudo code, have a look for better understanding

C++

 `// C++ program to check whether number has` `// exactly four distinct factors or not` `#include ` `using` `namespace` `std;`   `// Initialize global variable according` `// to given condition so that it can be` `// accessible to all function` `const` `int` `N = 1e6;` `bool` `fourDiv[N + 1];`   `// Function to calculate all number having` `// four distinct distinct factors` `void` `fourDistinctFactors()` `{` `    ``// Create a boolean array "prime[0..n]" and ` `    ``// initialize all entries it as true. A value ` `    ``// in prime[i] will finally be false if i is ` `    ``// not a prime, else true.` `    ``bool` `primeAll[N + 1];` `    ``memset``(primeAll, ``true``, ``sizeof``(primeAll));`   `    ``for` `(``int` `p = 2; p * p <= N; p++) {`   `        ``// If prime[p] is not changed, then it` `        ``// is a prime` `        ``if` `(primeAll[p] == ``true``) {`   `            ``// Update all multiples of p` `            ``for` `(``int` `i = p * 2; i <= N; i += p)` `                ``primeAll[i] = ``false``;` `        ``}` `    ``}`   `    ``// Initialize prime[] array which will` `    ``// contains all the primes from 1-N` `    ``vector<``int``> prime;`   `    ``for` `(``int` `p = 2; p <= N; p++)` `        ``if` `(primeAll[p])` `            ``prime.push_back(p);`   `    ``// Set the marking of all primes to false` `    ``memset``(fourDiv, ``false``, ``sizeof``(fourDiv));`   `    ``// Iterate over all the prime numbers` `    ``for` `(``int` `i = 0; i < prime.size(); ++i) {` `        ``int` `p = prime[i];`   `        ``// Mark cube root of prime numbers` `        ``if` `(1LL * p * p * p <= N)` `            ``fourDiv[p * p * p] = ``true``;`   `        ``for` `(``int` `j = i + 1; j < prime.size(); ++j) {` `            ``int` `q = prime[j];`   `            ``if` `(1LL * p * q > N)` `                ``break``;`   `            ``// Mark product of prime numbers` `            ``fourDiv[p * q] = ``true``;` `        ``}` `    ``}` `}`   `// Driver program` `int` `main()` `{` `    ``fourDistinctFactors();`   `    ``int` `num = 10;` `    ``if` `(fourDiv[num])` `        ``cout << ``"Yes\n"``;` `    ``else` `        ``cout << ``"No\n"``;`   `    ``num = 12;` `    ``if` `(fourDiv[num])` `        ``cout << ``"Yes\n"``;` `    ``else` `        ``cout << ``"No\n"``;`   `    ``return` `0;` `}`

Java

 `// Java program to check whether number has` `// exactly four distinct factors or not` `import` `java.util.*;` `class` `GFG{` `// Initialize global variable according` `// to given condition so that it can be` `// accessible to all function` `static` `int` `N = (``int``)1E6;` `static` `boolean``[] fourDiv=``new` `boolean``[N + ``1``];`   `// Function to calculate all number having` `// four distinct distinct factors` `static` `void` `fourDistinctFactors()` `{` `    ``// Create a boolean array "prime[0..n]" and ` `    ``// initialize all entries it as true. A value ` `    ``// in prime[i] will finally be false if i is ` `    ``// not a prime, else true.` `    ``boolean``[] primeAll=``new` `boolean``[N + ``1``];`   `    ``for` `(``int` `p = ``2``; p * p <= N; p++) {`   `        ``// If prime[p] is not changed, then it` `        ``// is a prime` `        ``if` `(primeAll[p] == ``false``) {`   `            ``// Update all multiples of p` `            ``for` `(``int` `i = p * ``2``; i <= N; i += p)` `                ``primeAll[i] = ``true``;` `        ``}` `    ``}`   `    ``// Initialize prime[] array which will` `    ``// contains all the primes from 1-N` `    ``ArrayList prime=``new` `ArrayList();`   `    ``for` `(``int` `p = ``2``; p <= N; p++)` `        ``if` `(!primeAll[p])` `            ``prime.add(p);`     `    ``// Iterate over all the prime numbers` `    ``for` `(``int` `i = ``0``; i < prime.size(); ++i) {` `        ``int` `p = prime.get(i);`   `        ``// Mark cube root of prime numbers` `        ``if` `(1L * p * p * p <= N)` `            ``fourDiv[p * p * p] = ``true``;`   `        ``for` `(``int` `j = i + ``1``; j < prime.size(); ++j) {` `            ``int` `q = prime.get(j);`   `            ``if` `(1L * p * q > N)` `                ``break``;`   `            ``// Mark product of prime numbers` `            ``fourDiv[p * q] = ``true``;` `        ``}` `    ``}` `}`   `// Driver program` `public` `static` `void` `main(String[] args)` `{` `    ``fourDistinctFactors();`   `    ``int` `num = ``10``;` `    ``if` `(fourDiv[num])` `        ``System.out.println(``"Yes"``);` `    ``else` `        ``System.out.println(``"No"``);`   `    ``num = ``12``;` `    ``if` `(fourDiv[num])` `        ``System.out.println(``"Yes"``);` `    ``else` `        ``System.out.println(``"No"``);`   `}` `}` `// This code is contributed by mits`

Python3

 `# Python3 program to check whether number ` `# has exactly four distinct factors or not`   `# Initialize global variable according to ` `# given condition so that it can be ` `# accessible to all function` `N ``=` `1000001``;` `fourDiv ``=` `[``False``] ``*` `(N ``+` `1``);`   `# Function to calculate all number ` `# having four distinct factors` `def` `fourDistinctFactors():` `    `  `    ``# Create a boolean array "prime[0..n]" ` `    ``# and initialize all entries it as true. ` `    ``# A value in prime[i] will finally be ` `    ``# false if i is not a prime, else true.` `    ``primeAll ``=` `[``True``] ``*` `(N ``+` `1``);` `    ``p ``=` `2``;`   `    ``while` `(p ``*` `p <``=` `N): `   `        ``# If prime[p] is not changed, then it ` `        ``# is a prime` `        ``if` `(primeAll[p] ``=``=` `True``):`   `            ``# Update all multiples of p` `            ``i ``=` `p ``*` `2``;` `            ``while` `(i <``=` `N):` `                ``primeAll[i] ``=` `False``;` `                ``i ``+``=` `p;` `        ``p ``+``=` `1``;`   `    ``# Initialize prime[] array which will ` `    ``# contain all the primes from 1-N` `    ``prime ``=` `[];`   `    ``for` `p ``in` `range``(``2``, N ``+` `1``):` `        ``if` `(primeAll[p]):` `            ``prime.append(p);`   `    ``# Iterate over all the prime numbers` `    ``for` `i ``in` `range``(``len``(prime)): ` `        ``p ``=` `prime[i];`   `        ``# Mark cube root of prime numbers` `        ``if` `(``1` `*` `p ``*` `p ``*` `p <``=` `N):` `            ``fourDiv[p ``*` `p ``*` `p] ``=` `True``;`   `        ``for` `j ``in` `range``(i ``+` `1``, ``len``(prime)):` `            ``q ``=` `prime[j];`   `            ``if` `(``1` `*` `p ``*` `q > N):` `                ``break``;`   `            ``# Mark product of prime numbers` `            ``fourDiv[p ``*` `q] ``=` `True``;`   `# Driver Code` `fourDistinctFactors();`   `num ``=` `10``;` `if` `(fourDiv[num]):` `    ``print``(``"Yes"``);` `else``:` `    ``print``(``"No"``);`   `num ``=` `12``;` `if` `(fourDiv[num]):` `    ``print``(``"Yes"``);` `else``:` `    ``print``(``"No"``);`   `# This code is contributed by mits`

C#

 `// C# program to check whether number has` `// exactly four distinct factors or not` `using` `System;` `using` `System.Collections;`   `class` `GFG` `{` `    `  `// Initialize global variable according` `// to given condition so that it can be` `// accessible to all function` `static` `int` `N = (``int``)1E6;` `static` `bool``[] fourDiv = ``new` `bool``[N + 1];`   `// Function to calculate all number having` `// four distinct distinct factors` `static` `void` `fourDistinctFactors()` `{` `    ``// Create a boolean array "prime[0..n]" and ` `    ``// initialize all entries it as true. A value ` `    ``// in prime[i] will finally be false if i is ` `    ``// not a prime, else true.` `    ``bool``[] primeAll = ``new` `bool``[N + 1];`   `    ``for` `(``int` `p = 2; p * p <= N; p++) ` `    ``{`   `        ``// If prime[p] is not changed, ` `        ``// then it is a prime` `        ``if` `(primeAll[p] == ``false``) ` `        ``{`   `            ``// Update all multiples of p` `            ``for` `(``int` `i = p * 2; i <= N; i += p)` `                ``primeAll[i] = ``true``;` `        ``}` `    ``}`   `    ``// Initialize prime[] array which will` `    ``// contains all the primes from 1-N` `    ``ArrayList prime = ``new` `ArrayList();`   `    ``for` `(``int` `p = 2; p <= N; p++)` `        ``if` `(!primeAll[p])` `            ``prime.Add(p);`   `    ``// Iterate over all the prime numbers` `    ``for` `(``int` `i = 0; i < prime.Count; ++i)` `    ``{` `        ``int` `p = (``int``)prime[i];`   `        ``// Mark cube root of prime numbers` `        ``if` `(1L * p * p * p <= N)` `            ``fourDiv[p * p * p] = ``true``;`   `        ``for` `(``int` `j = i + 1; j < prime.Count; ++j) ` `        ``{` `            ``int` `q = (``int``)prime[j];`   `            ``if` `(1L * p * q > N)` `                ``break``;`   `            ``// Mark product of prime numbers` `            ``fourDiv[p * q] = ``true``;` `        ``}` `    ``}` `}`   `// Driver Code` `public` `static` `void` `Main()` `{` `    ``fourDistinctFactors();`   `    ``int` `num = 10;` `    ``if` `(fourDiv[num])` `        ``Console.WriteLine(``"Yes"``);` `    ``else` `        ``Console.WriteLine(``"No"``);`   `    ``num = 12;` `    ``if` `(fourDiv[num])` `        ``Console.WriteLine(``"Yes"``);` `    ``else` `        ``Console.WriteLine(``"No"``);` `}` `}`   `// This code is contributed by mits`

PHP

 ` ``\$N``)` `                ``break``;`   `            ``// Mark product of ` `            ``// prime numbers` `            ``\$fourDiv``[``\$p` `* ``\$q``] = true;` `        ``}` `    ``}` `}`   `// Driver Code` `fourDistinctFactors();`   `\$num` `= 10;` `if` `(``\$fourDiv``[``\$num``])` `    ``echo` `"Yes\n"``;` `else` `    ``echo` `"No\n"``;`   `\$num` `= 12;` `if` `(``\$fourDiv``[``\$num``])` `    ``echo` `"Yes\n"``;` `else` `    ``echo` `"No\n"``;`   `// This code is contributed by mits` `?>`

Javascript

 ``

Output:

```Yes
No```

Time Complexity: O(n log(log n))
Auxiliary Space: O(n)