# Check for an array element that is co-prime with all others

Given an array arr[] of positive integers where 2 ≤ arr[i] ≤ 106 for all possible values of i. The task is to check whether there exists at least one element in the given array that forms co-prime pair with all other elements of the array. If no such element exists then print No else print Yes.

Examples:

Input: arr[] = {2, 8, 4, 10, 6, 7}
Output: Yes
7 is co-prime with all the other elements of the array

Input: arr[] = {3, 6, 9, 12}
Output: No

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive approach: A simple solution is to check whether the gcd of every element with all other elements is equal to 1. Time complexity of this solution is O(n2).

Efficient approach: An efficient solution is to generate all the prime factors of integers in the given array. Using hash, store the count of every element which is a prime factor of any of the number in the array. If the element does not contain any common prime factor with other elements, it always forms a co-prime pair with other elements.
For generating prime factors please go through the article Prime Factorization using Sieve in O(log n)

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `#define MAXN 1000001 ` ` `  `// Stores smallest prime factor for every number ` `int` `spf[MAXN]; ` ` `  `// Hash to store prime factors count ` `int` `hash1[MAXN] = { 0 }; ` ` `  `// Function to calculate SPF (Smallest Prime Factor) ` `// for every number till MAXN ` `void` `sieve() ` `{ ` `    ``spf = 1; ` `    ``for` `(``int` `i = 2; i < MAXN; i++) ` ` `  `        ``// Marking smallest prime factor for every ` `        ``// number to be itself ` `        ``spf[i] = i; ` ` `  `    ``// Separately marking spf for every even ` `    ``// number as 2 ` `    ``for` `(``int` `i = 4; i < MAXN; i += 2) ` `        ``spf[i] = 2; ` ` `  `    ``// Checking if i is prime ` `    ``for` `(``int` `i = 3; i * i < MAXN; i++) { ` ` `  `        ``// Marking SPF for all numbers divisible by i ` `        ``if` `(spf[i] == i) { ` `            ``for` `(``int` `j = i * i; j < MAXN; j += i) ` ` `  `                ``// Marking spf[j] if it is not ` `                ``// previously marked ` `                ``if` `(spf[j] == j) ` `                    ``spf[j] = i; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to store the prime factors after dividing ` `// by the smallest prime factor at every step ` `void` `getFactorization(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != 1) { ` `        ``temp = spf[x]; ` `        ``if` `(x % temp == 0) { ` ` `  `            ``// Storing the count of ` `            ``// prime factors in hash ` `            ``hash1[spf[x]]++; ` `            ``x = x / spf[x]; ` `        ``} ` `        ``while` `(x % temp == 0) ` `            ``x = x / temp; ` `    ``} ` `} ` ` `  `// Function that returns true if there are ` `// no common prime factors between x ` `// and other numbers of the array ` `bool` `check(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != 1) { ` `        ``temp = spf[x]; ` ` `  `        ``// Checking whether it common ` `        ``// prime factor with other numbers ` `        ``if` `(x % temp == 0 && hash1[temp] > 1) ` `            ``return` `false``; ` `        ``while` `(x % temp == 0) ` `            ``x = x / temp; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function that returns true if there is ` `// an element in the array which is coprime ` `// with all the other elements of the array ` `bool` `hasValidNum(``int` `arr[], ``int` `n) ` `{ ` ` `  `    ``// Using sieve for generating prime factors ` `    ``sieve(); ` ` `  `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``getFactorization(arr[i]); ` ` `  `    ``// Checking the common prime factors ` `    ``// with other numbers ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``if` `(check(arr[i])) ` `            ``return` `true``; ` ` `  `    ``return` `false``; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 2, 8, 4, 10, 6, 7 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``if` `(hasValidNum(arr, n)) ` `        ``cout << ``"Yes"``; ` `    ``else` `        ``cout << ``"No"``; ` `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach ` `class` `GFG ` `{ ` `     `  `static` `int` `MAXN = ``1000001``; ` ` `  `// Stores smallest prime factor for every number ` `static` `int``[] spf = ``new` `int``[MAXN]; ` ` `  `// Hash to store prime factors count ` `static` `int``[] hash1 = ``new` `int``[MAXN]; ` ` `  `// Function to calculate SPF (Smallest Prime Factor) ` `// for every number till MAXN ` `static` `void` `sieve() ` `{ ` `    ``spf[``1``] = ``1``; ` `    ``for` `(``int` `i = ``2``; i < MAXN; i++) ` ` `  `        ``// Marking smallest prime factor for every ` `        ``// number to be itself ` `        ``spf[i] = i; ` ` `  `    ``// Separately marking spf for every even ` `    ``// number as 2 ` `    ``for` `(``int` `i = ``4``; i < MAXN; i += ``2``) ` `        ``spf[i] = ``2``; ` ` `  `    ``// Checking if i is prime ` `    ``for` `(``int` `i = ``3``; i * i < MAXN; i++)  ` `    ``{ ` ` `  `        ``// Marking SPF for all numbers divisible by i ` `        ``if` `(spf[i] == i)  ` `        ``{ ` `            ``for` `(``int` `j = i * i; j < MAXN; j += i) ` ` `  `                ``// Marking spf[j] if it is not ` `                ``// previously marked ` `                ``if` `(spf[j] == j) ` `                    ``spf[j] = i; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to store the prime factors after dividing ` `// by the smallest prime factor at every step ` `static` `void` `getFactorization(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != ``1``)  ` `    ``{ ` `        ``temp = spf[x]; ` `        ``if` `(x % temp == ``0``)  ` `        ``{ ` ` `  `            ``// Storing the count of ` `            ``// prime factors in hash ` `            ``hash1[spf[x]]++; ` `            ``x = x / spf[x]; ` `        ``} ` `        ``while` `(x % temp == ``0``) ` `            ``x = x / temp; ` `    ``} ` `} ` ` `  `// Function that returns true if there are ` `// no common prime factors between x ` `// and other numbers of the array ` `static` `boolean` `check(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != ``1``)  ` `    ``{ ` `        ``temp = spf[x]; ` ` `  `        ``// Checking whether it common ` `        ``// prime factor with other numbers ` `        ``if` `(x % temp == ``0` `&& hash1[temp] > ``1``) ` `            ``return` `false``; ` `        ``while` `(x % temp == ``0``) ` `            ``x = x / temp; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function that returns true if there is ` `// an element in the array which is coprime ` `// with all the other elements of the array ` `static` `boolean` `hasValidNum(``int` `[]arr, ``int` `n) ` `{ ` ` `  `    ``// Using sieve for generating prime factors ` `    ``sieve(); ` ` `  `    ``for` `(``int` `i = ``0``; i < n; i++) ` `        ``getFactorization(arr[i]); ` ` `  `    ``// Checking the common prime factors ` `    ``// with other numbers ` `    ``for` `(``int` `i = ``0``; i < n; i++) ` `        ``if` `(check(arr[i])) ` `            ``return` `true``; ` ` `  `    ``return` `false``; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main (String[] args)  ` `{ ` ` `  `    ``int` `[]arr = { ``2``, ``8``, ``4``, ``10``, ``6``, ``7` `}; ` `    ``int` `n = arr.length; ` ` `  `    ``if` `(hasValidNum(arr, n)) ` `        ``System.out.println(``"Yes"``); ` `    ``else` `        ``System.out.println(``"No"``); ` `} ` `} ` ` `  `// This code is contributed by chandan_jnu `

## Python3

 `# Python3 implementation of the approach ` `MAXN ``=` `1000001` ` `  `# Stores smallest prime factor for  ` `# every number ` `spf ``=` `[i ``for` `i ``in` `range``(MAXN)] ` ` `  `# Hash to store prime factors count ` `hash1 ``=` `[``0` `for` `i ``in` `range``(MAXN)] ` ` `  `# Function to calculate SPF (Smallest  ` `# Prime Factor) for every number till MAXN ` `def` `sieve(): ` ` `  `    ``# Separately marking spf for  ` `    ``# every even number as 2 ` `    ``for` `i ``in` `range``(``4``, MAXN, ``2``): ` `        ``spf[i] ``=` `2` ` `  `    ``# Checking if i is prime ` `    ``for` `i ``in` `range``(``3``, MAXN): ` ` `  `        ``if` `i ``*` `i < MAXN: ` `            ``break` ` `  `        ``# Marking SPF for all numbers ` `        ``# divisible by i ` `        ``if` `(spf[i] ``=``=` `i): ` `            ``for` `j ``in` `range``(i ``*` `i, MAXN, i): ` ` `  `                ``# Marking spf[j] if it is not ` `                ``# previously marked ` `                ``if` `(spf[j] ``=``=` `j): ` `                    ``spf[j] ``=` `i ` ` `  `# Function to store the prime factors  ` `# after dividing by the smallest prime  ` `# factor at every step ` `def` `getFactorization(x): ` ` `  `    ``while` `(x !``=` `1``): ` `        ``temp ``=` `spf[x] ` `        ``if` `(x ``%` `temp ``=``=` `0``): ` ` `  `            ``# Storing the count of ` `            ``# prime factors in hash ` `            ``hash1[spf[x]] ``+``=` `1` `            ``x ``=` `x ``/``/` `spf[x] ` ` `  `        ``while` `(x ``%` `temp ``=``=` `0``): ` `            ``x ``=` `x ``/``/` `temp ` ` `  `# Function that returns true if there  ` `# are no common prime factors between x ` `# and other numbers of the array ` `def` `check(x): ` ` `  `    ``while` `(x !``=` `1``): ` `        ``temp ``=` `spf[x] ` ` `  `        ``# Checking whether it common ` `        ``# prime factor with other numbers ` `        ``if` `(x ``%` `temp ``=``=` `0` `and` `hash1[temp] > ``1``): ` `            ``return` `False` `        ``while` `(x ``%` `temp ``=``=` `0``): ` `            ``x ``=` `x ``/``/``temp ` `     `  `    ``return` `True` ` `  `# Function that returns true if there is ` `# an element in the array which is coprime ` `# with all the other elements of the array ` `def` `hasValidNum(arr, n): ` ` `  `    ``# Using sieve for generating  ` `    ``# prime factors ` `    ``sieve() ` ` `  `    ``for` `i ``in` `range``(n): ` `        ``getFactorization(arr[i]) ` ` `  `    ``# Checking the common prime factors ` `    ``# with other numbers ` `    ``for` `i ``in` `range``(n): ` `        ``if` `(check(arr[i])): ` `            ``return` `True` ` `  `    ``return` `False` ` `  `# Driver code ` `arr ``=` `[``2``, ``8``, ``4``, ``10``, ``6``, ``7``] ` `n ``=` `len``(arr) ` ` `  `if` `(hasValidNum(arr, n)): ` `    ``print``(``"Yes"``) ` `else``: ` `    ``print``(``"No"``) ` ` `  `# This code is contributed by mohit kumar `

## C#

 `// C# implementation of the approach ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `static` `int` `MAXN=1000001; ` ` `  `// Stores smallest prime factor for every number ` `static` `int``[] spf = ``new` `int``[MAXN]; ` ` `  `// Hash to store prime factors count ` `static` `int``[] hash1 = ``new` `int``[MAXN]; ` ` `  `// Function to calculate SPF (Smallest Prime Factor) ` `// for every number till MAXN ` `static` `void` `sieve() ` `{ ` `    ``spf = 1; ` `    ``for` `(``int` `i = 2; i < MAXN; i++) ` ` `  `        ``// Marking smallest prime factor for every ` `        ``// number to be itself ` `        ``spf[i] = i; ` ` `  `    ``// Separately marking spf for every even ` `    ``// number as 2 ` `    ``for` `(``int` `i = 4; i < MAXN; i += 2) ` `        ``spf[i] = 2; ` ` `  `    ``// Checking if i is prime ` `    ``for` `(``int` `i = 3; i * i < MAXN; i++)  ` `    ``{ ` ` `  `        ``// Marking SPF for all numbers divisible by i ` `        ``if` `(spf[i] == i)  ` `        ``{ ` `            ``for` `(``int` `j = i * i; j < MAXN; j += i) ` ` `  `                ``// Marking spf[j] if it is not ` `                ``// previously marked ` `                ``if` `(spf[j] == j) ` `                    ``spf[j] = i; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to store the prime factors after dividing ` `// by the smallest prime factor at every step ` `static` `void` `getFactorization(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != 1)  ` `    ``{ ` `        ``temp = spf[x]; ` `        ``if` `(x % temp == 0)  ` `        ``{ ` ` `  `            ``// Storing the count of ` `            ``// prime factors in hash ` `            ``hash1[spf[x]]++; ` `            ``x = x / spf[x]; ` `        ``} ` `        ``while` `(x % temp == 0) ` `            ``x = x / temp; ` `    ``} ` `} ` ` `  `// Function that returns true if there are ` `// no common prime factors between x ` `// and other numbers of the array ` `static` `bool` `check(``int` `x) ` `{ ` `    ``int` `temp; ` `    ``while` `(x != 1)  ` `    ``{ ` `        ``temp = spf[x]; ` ` `  `        ``// Checking whether it common ` `        ``// prime factor with other numbers ` `        ``if` `(x % temp == 0 && hash1[temp] > 1) ` `            ``return` `false``; ` `        ``while` `(x % temp == 0) ` `            ``x = x / temp; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function that returns true if there is ` `// an element in the array which is coprime ` `// with all the other elements of the array ` `static` `bool` `hasValidNum(``int` `[]arr, ``int` `n) ` `{ ` ` `  `    ``// Using sieve for generating prime factors ` `    ``sieve(); ` ` `  `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``getFactorization(arr[i]); ` ` `  `    ``// Checking the common prime factors ` `    ``// with other numbers ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``if` `(check(arr[i])) ` `            ``return` `true``; ` ` `  `    ``return` `false``; ` `} ` ` `  `// Driver code ` `static` `void` `Main() ` `{ ` `    ``int` `[]arr = { 2, 8, 4, 10, 6, 7 }; ` `    ``int` `n = arr.Length; ` ` `  `    ``if` `(hasValidNum(arr, n)) ` `        ``Console.WriteLine(``"Yes"``); ` `    ``else` `        ``Console.WriteLine(``"No"``); ` `} ` `} ` ` `  `// This code is contributed by chandan_jnu `

## PHP

 ` 1) ` `            ``return` `false; ` `        ``while` `(``\$x` `% ``\$temp` `== 0) ` `            ``\$x` `= (int)(``\$x` `/ ``\$temp``); ` `    ``} ` `    ``return` `true; ` `} ` ` `  `// Function that returns true if there is ` `// an element in the array which is coprime ` `// with all the other elements of the array ` `function` `hasValidNum(``\$arr``, ``\$n``) ` `{ ` `    ``global` `\$spf``,``\$MAXN``,``\$hash1``; ` ` `  `    ``// Using sieve for generating prime factors ` `    ``sieve(); ` ` `  `    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) ` `        ``getFactorization(``\$arr``[``\$i``]); ` ` `  `    ``// Checking the common prime factors ` `    ``// with other numbers ` `    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) ` `        ``if` `(check(``\$arr``[``\$i``])) ` `            ``return` `true; ` ` `  `    ``return` `false; ` `} ` ` `  `// Driver code ` `    ``\$arr` `= ``array``( 2, 8, 4, 10, 6, 7 ); ` `    ``\$n` `= ``count``(``\$arr``); ` ` `  `    ``if` `(hasValidNum(``\$arr``, ``\$n``)) ` `        ``echo` `"Yes"``; ` `    ``else` `        ``echo` `"No"``; ` ` `  `// This code is contributed by chandan_jnu ` `?> `

Output:

```Yes
```

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