# Convert the array such that the GCD of the array becomes 1

Given an array of positive elements and a positive integer k, the task is to convert the GCD of the array to 1. To make it possible, only one operation is allowed any number of times i.e. choose any element of the array & divide with a number d where d <= k.

Examples:

Input : arr = {10, 15, 30}, k = 6
Output : Yes
Divide all the elements of array by 5 as it is
the divisor of all elements and also less than k i.e. 6.
It gives as the sequence like {2, 3, 6}. Now, divide
2 by 2, 3 by 3 and 6 by 2. Then, the sequence
becomes {1, 1, 3}. Now, the gcd of the array is 1.

Input : arr = {5, 10, 20}, k = 4
Output : No
Here, divide 10 with 2 the sequence becomes
{5, 5, 20}. Then, divide 20 by 2 and again by 2.
Finally, the sequence becomes {5, 5, 5}. To make
the gcd of this sequence 1, divide each element
by 5. But 5 > 4 so here it is impossible to make the
gcd 1.

Approach:

• If there exists a positive prime number greater than k that divides each element of the array then the answer is No.
• If the largest prime factor of the GCD of the array is less than or equal to k then the answer is Yes.
• First, find the GCD of the array then check if there exists a prime factor of the GCD that is greater than k.
• For this, calculate largest prime factor of GCD.

## C++

 `// C++ program to check if it is ` `// possible to convert the gcd of ` `// the array to 1 by applying the ` `// given operation ` `#include ` `using` `namespace` `std; ` ` `  `// Function to get gcd of the array. ` `int` `getGcd(``int``* arr, ``int` `n) ` `{ ` `    ``int` `gcd = arr; ` `    ``for` `(``int` `i = 1; i < n; i++) ` `        ``gcd = __gcd(arr[i], gcd); ` ` `  `    ``return` `gcd; ` `} ` ` `  `// Function to check if it is possible. ` `bool` `convertGcd(``int``* arr, ``int` `n, ``int` `k) ` `{ ` `    ``// Getting the gcd of array ` `    ``int` `gcd = getGcd(arr, n); ` ` `  `    ``// Initially taking max_prime factor is 1. ` `    ``int` `max_prime = 1; ` ` `  `    ``// find maximum of all the prime factors ` `    ``// till sqrt(gcd). ` `    ``for` `(``int` `i = 2; i <= ``sqrt``(gcd); i++) { ` `        ``while` `(gcd % i == 0) { ` `            ``gcd /= i; ` `            ``max_prime = max(max_prime, i); ` `        ``} ` `    ``} ` ` `  `    ``// either GCD is reduced to 1 or a prime factor ` `    ``// greater than sqrt(gcd) ` `    ``max_prime = max(max_prime, gcd); ` ` `  `    ``return` `(max_prime <= k); ` `} ` ` `  `// Drivers code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 10, 15, 30 }; ` `    ``int` `k = 6; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``if` `(convertGcd(arr, n, k) == ``true``) ` `       ``cout << ``"Yes"``; ` `    ``else` `       ``cout << ``"No"``; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to check if it is ` `// possible to convert the gcd of ` `// the array to 1 by applying the ` `// given operation ` `import` `java.io.*; ` ` `  `class` `GFG { ` ` `  `    ``// Recursive function to return ` `    ``// gcd of a and b ` `    ``static` `int` `__gcd(``int` `a, ``int` `b) ` `    ``{ ` `        ``// Everything divides 0  ` `        ``if` `(a == ``0` `|| b == ``0``) ` `            ``return` `0``; ` `     `  `        ``// base case ` `        ``if` `(a == b) ` `            ``return` `a; ` `     `  `        ``// a is greater ` `        ``if` `(a > b) ` `            ``return` `__gcd(a-b, b); ` `        ``return` `__gcd(a, b-a); ` `    ``} ` `     `  `    ``// Function to get gcd of the array. ` `    ``static` `int` `getGcd(``int` `arr[], ``int` `n) ` `    ``{ ` `        ``int` `gcd = arr[``0``]; ` `        ``for` `(``int` `i = ``1``; i < n; i++) ` `            ``gcd = __gcd(arr[i], gcd); ` `     `  `        ``return` `gcd; ` `    ``} ` ` `  `    ``// Function to check if it is possible. ` `    ``static` `boolean` `convertGcd(``int` `[]arr,  ` `                             ``int` `n, ``int` `k) ` `    ``{ ` `        ``// Getting the gcd of array ` `        ``int` `gcd = getGcd(arr, n); ` `     `  `        ``// Initially taking max_prime ` `        ``// factor is 1. ` `        ``int` `max_prime = ``1``; ` `     `  `        ``// find maximum of all the prime  ` `        ``// factors till sqrt(gcd). ` `        ``for` `(``int` `i = ``2``; i <= Math.sqrt(gcd); ` `                                        ``i++) ` `        ``{ ` `            ``while` `(gcd % i == ``0``) { ` `                ``gcd /= i; ` `                ``max_prime =  ` `                     ``Math.max(max_prime, i); ` `            ``} ` `        ``} ` `     `  `        ``// either GCD is reduced to 1 or a ` `        ``// prime factor greater than sqrt(gcd) ` `        ``max_prime = Math.max(max_prime, gcd); ` `     `  `        ``return` `(max_prime <= k); ` `    ``} ` ` `  `    ``// Drivers code ` `    ``public` `static` `void` `main (String[] args) ` `    ``{ ` `        ``int` `[]arr = { ``10``, ``15``, ``30` `}; ` `        ``int` `k = ``6``; ` `        ``int` `n = arr.length; ` `     `  `        ``if` `(convertGcd(arr, n, k) == ``true``) ` `            ``System.out.println( ``"Yes"``); ` `        ``else` `            ``System.out.println( ``"No"``); ` `    ``} ` `} ` ` `  `// This code is contributed by anuj_67. `

## Python3

 `# Python 3 program to check if it is ` `# possible to convert the gcd of ` `# the array to 1 by applying the ` `# given operation ` `from` `math ``import` `gcd as __gcd, sqrt ` ` `  `# Function to get gcd of the array. ` `def` `getGcd(arr, n): ` `    ``gcd ``=` `arr[``0``]; ` `    ``for` `i ``in` `range``(``1``, n, ``1``): ` `        ``gcd ``=` `__gcd(arr[i], gcd) ` ` `  `    ``return` `gcd ` ` `  `# Function to check if it is possible. ` `def` `convertGcd(arr, n, k): ` `     `  `    ``# Getting the gcd of array ` `    ``gcd ``=` `getGcd(arr, n) ` ` `  `    ``# Initially taking max_prime  ` `    ``# factor is 1. ` `    ``max_prime ``=` `1` ` `  `    ``# find maximum of all the  ` `    ``# prime factors till sqrt(gcd) ` `    ``p ``=` `int``(sqrt(gcd)) ``+` `1` `    ``for` `i ``in` `range``(``2``, p, ``1``): ` `        ``while` `(gcd ``%` `i ``=``=` `0``): ` `            ``gcd ``=` `int``(gcd ``/` `i) ` `            ``max_prime ``=` `max``(max_prime, i) ` ` `  `    ``# either GCD is reduced to 1 or a  ` `    ``# prime factor greater than sqrt(gcd) ` `    ``max_prime ``=` `max``(max_prime, gcd) ` ` `  `    ``return` `(max_prime <``=` `k) ` ` `  `# Drivers code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``arr ``=` `[``10``, ``15``, ``30``] ` `    ``k ``=` `6` `    ``n ``=` `len``(arr) ` ` `  `    ``if` `(convertGcd(arr, n, k) ``=``=` `True``): ` `        ``print``(``"Yes"``) ` `    ``else``: ` `        ``print``(``"No"``) ` ` `  `# This code is contributed by ` `# Sahil_Shelangia `

## C#

 `// C# program to check if it is ` `// possible to convert the gcd of ` `// the array to 1 by applying the ` `// given operation ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Recursive function to return ` `    ``// gcd of a and b ` `    ``static` `int` `__gcd(``int` `a, ``int` `b) ` `    ``{ ` `        ``// Everything divides 0  ` `        ``if` `(a == 0 || b == 0) ` `            ``return` `0; ` `     `  `        ``// base case ` `        ``if` `(a == b) ` `            ``return` `a; ` `     `  `        ``// a is greater ` `        ``if` `(a > b) ` `            ``return` `__gcd(a-b, b); ` `        ``return` `__gcd(a, b-a); ` `    ``} ` `     `  `    ``// Function to get gcd of the array. ` `    ``static` `int` `getGcd(``int` `[]arr, ``int` `n) ` `    ``{ ` `        ``int` `gcd = arr; ` `        ``for` `(``int` `i = 1; i < n; i++) ` `            ``gcd = __gcd(arr[i], gcd); ` `     `  `        ``return` `gcd; ` `    ``} ` ` `  `    ``// Function to check if it is possible. ` `    ``static` `bool` `convertGcd(``int` `[]arr,  ` `                            ``int` `n, ``int` `k) ` `    ``{ ` `        ``// Getting the gcd of array ` `        ``int` `gcd = getGcd(arr, n); ` `     `  `        ``// Initially taking max_prime ` `        ``// factor is 1. ` `        ``int` `max_prime = 1; ` `     `  `        ``// find maximum of all the prime  ` `        ``// factors till sqrt(gcd). ` `        ``for` `(``int` `i = 2; i <= Math.Sqrt(gcd); ` `                                        ``i++) ` `        ``{ ` `            ``while` `(gcd % i == 0) { ` `                ``gcd /= i; ` `                ``max_prime =  ` `                    ``Math.Max(max_prime, i); ` `            ``} ` `        ``} ` `     `  `        ``// either GCD is reduced to 1 or a ` `        ``// prime factor greater than sqrt(gcd) ` `        ``max_prime = Math.Max(max_prime, gcd); ` `     `  `        ``return` `(max_prime <= k); ` `    ``} ` ` `  `    ``// Drivers code ` `    ``public` `static` `void` `Main () ` `    ``{ ` `        ``int` `[]arr = { 10, 15, 30 }; ` `        ``int` `k = 6; ` `        ``int` `n = arr.Length; ` `     `  `        ``if` `(convertGcd(arr, n, k) == ``true``) ` `            ``Console.WriteLine( ``"Yes"``); ` `        ``else` `            ``Console.WriteLine( ``"No"``); ` `    ``} ` `} ` ` `  `// This code is contributed by anuj_67. `

## PHP

 ` ``\$b``) ` `        ``return` `__gcd(``\$a` `- ``\$b` `, ``\$b``) ; ` ` `  `    ``return` `__gcd( ``\$a` `, ``\$b` `- ``\$a``) ; ` `} ` ` `  `// Function to get gcd of the array. ` `function` `getGcd(``\$arr``, ``\$n``) ` `{ ` `    ``\$gcd` `= ``\$arr``; ` `    ``for` `(``\$i` `= 1; ``\$i` `< ``\$n``; ``\$i``++) ` `        ``\$gcd` `= __gcd(``\$arr``[``\$i``], ``\$gcd``); ` ` `  `    ``return` `\$gcd``; ` `} ` ` `  `// Function to check if it is possible. ` `function` `convertGcd( ``\$arr``, ``\$n``, ``\$k``) ` `{ ` `     `  `    ``// Getting the gcd of array ` `    ``\$gcd` `= getGcd(``\$arr``, ``\$n``); ` ` `  `    ``// Initially taking max_prime ` `    ``// factor is 1. ` `    ``\$max_prime` `= 1; ` ` `  `    ``// find maximum of all the prime ` `    ``// factors till sqrt(gcd). ` `    ``for``(``\$i` `= 2; ``\$i` `<= sqrt(``\$gcd``); ``\$i``++) ` `    ``{ ` `        ``while` `(``\$gcd` `% ``\$i` `== 0) ` `        ``{ ` `            ``\$gcd` `/= ``\$i``; ` `            ``\$max_prime` `= max(``\$max_prime``, ``\$i``); ` `        ``} ` `    ``} ` ` `  `    ``// either GCD is reduced  ` `    ``// to 1 or a prime factor ` `    ``// greater than sqrt(gcd) ` `    ``\$max_prime` `= max(``\$max_prime``, ``\$gcd``); ` ` `  `    ``return` `(``\$max_prime` `<= ``\$k``); ` `} ` ` `  `    ``// Driver Code ` `    ``\$arr` `= ``array``(10, 15, 30); ` `    ``\$k` `= 6; ` `    ``\$n` `= ``count``(``\$arr``); ` ` `  `    ``if` `(convertGcd(``\$arr``, ``\$n``, ``\$k``) == true) ` `        ``echo` `"Yes"``; ` `    ``else` `        ``echo` `"No"``; ` ` `  `// This code is contributed by anuj_67. ` `?> `

Output:

```Yes
```

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