Smallest integer > 1 which divides every element of the given array

Last Updated : 22 Jun, 2022

Given an array arr[], the task is to find the smallest possible integer (other than 1) which divides every element of the given array.

Examples:

Input: arr[] = { 2, 4, 8 }
Output: 2
2 is the smallest possible number which divides the whole array.

Input: arr[] = { 4, 7, 5 }
Output: -1
There's no integer possible which divides the whole array other than 1.

Approach: We know that the GCD of the whole array will be the greatest integer that will divide every element of the array. If GCD = 1 then there's no integer possible that divides the whole array. However, if GCD > 1 then there exists integer(s) which divides the array completely. For example,

If GCD = 36 then
36 divides the whole array.
18 divides the whole array.
12 divides the whole array.
9 divides the whole array.
...
1 divides the whole array.
Thus, we see that all factors of 36 also divide the array. The smallest prime factor of 36 i.e. 2 is the smallest possible integer which divides the whole array. Hence, we need to find the smallest prime factor of the GCD as the required answer.

Below is the implementation of the above approach:

C++

// C++ implementation of the approach #include <bits/stdc++.h> using namespace std;  // Function to find the smallest divisor  int smallestDivisor(int x)  {      // if divisible by 2      if (x % 2 == 0)          return 2;         // iterate from 3 to sqrt(n)      for (int i = 3; i * i <= x; i += 2) {          if (x % i == 0)              return i;      }         return x;  }   // Function to return smallest possible integer // which divides the whole array int smallestInteger(int* arr, int n) {     // To store the GCD of all the array elements     int gcd = 0;     for (int i = 0; i < n; i++)         gcd = __gcd(gcd, arr[i]);      // Return the smallest prime factor     // of the gcd calculated     return smallestDivisor(gcd); }  // Driver code int main() {     int arr[] = { 2, 4, 8 };     int n = sizeof(arr) / sizeof(arr[0]);     cout << smallestInteger(arr, n);      return 0; }

Java

// Java implementation of the approach class GFG {  static int __gcd(int a, int b)  {      if (b == 0)          return a;      return __gcd(b, a % b);       }   // Function to find the smallest divisor  static int smallestDivisor(int x)  {      // if divisible by 2      if (x % 2 == 0)          return 2;       // iterate from 3 to sqrt(n)      for (int i = 3; i * i <= x; i += 2)      {          if (x % i == 0)              return i;      }       return x;  }   // Function to return smallest possible integer // which divides the whole array static int smallestInteger(int []arr, int n) {     // To store the GCD of all the array elements     int gcd = 0;     for (int i = 0; i < n; i++)         gcd = __gcd(gcd, arr[i]);      // Return the smallest prime factor     // of the gcd calculated     return smallestDivisor(gcd); }  // Driver code public static void main(String[] args) {     int []arr = { 2, 4, 8 };     int n = arr.length;     System.out.println(smallestInteger(arr, n)); } }  // This code is contributed by Code_Mech.

Python3

# Python3 implementation of the approach  from math import sqrt, gcd  # Function to find the smallest divisor  def smallestDivisor(x) :          # if divisible by 2      if (x % 2 == 0) :         return 2;           # iterate from 3 to sqrt(n)      for i in range(3, int(sqrt(x)) + 1, 2) :         if (x % i == 0) :             return i;           return x   # Function to return smallest possible  # integer which divides the whole array  def smallestInteger(arr, n) :          # To store the GCD of all the     # array elements      __gcd = 0;      for i in range(n) :         __gcd = gcd(__gcd, arr[i]);       # Return the smallest prime factor      # of the gcd calculated      return smallestDivisor(__gcd);   # Driver code  if __name__ == "__main__" :       arr = [ 2, 4, 8 ];     n = len(arr);          print(smallestInteger(arr, n));   # This code is contributed by Ryuga

// C# implementation of the approach using System;  class GFG {  static int __gcd(int a, int b)  {      if (b == 0)          return a;      return __gcd(b, a % b);       }   // Function to find the smallest divisor  static int smallestDivisor(int x)  {      // if divisible by 2      if (x % 2 == 0)          return 2;       // iterate from 3 to sqrt(n)      for (int i = 3; i * i <= x; i += 2)      {          if (x % i == 0)              return i;      }       return x;  }   // Function to return smallest possible integer // which divides the whole array static int smallestInteger(int []arr, int n) {     // To store the GCD of all the array elements     int gcd = 0;     for (int i = 0; i < n; i++)         gcd = __gcd(gcd, arr[i]);      // Return the smallest prime factor     // of the gcd calculated     return smallestDivisor(gcd); }  // Driver code static void Main() {     int []arr = { 2, 4, 8 };     int n = arr.Length;     Console.WriteLine(smallestInteger(arr, n)); } }  // This code is contributed by mits

PHP

<?php // PHP implementation of the approach function gcd($a, $b) {     // Everything divides 0     if($b == 0)         return $a;      return gcd($b , $a % $b); }  // Function to find the smallest divisor function smallestDivisor($x) {     // if divisible by 2     if ($x % 2 == 0)         return 2;      // iterate from 3 to sqrt(n)     for ($i = 3; $i < sqrt($x) + 1; $i += 2)     {         if ($x % $i == 0)             return $i;     }     return $x; }  // Function to return smallest possible // integer which divides the whole array function smallestInteger($arr, $n) {          // To store the GCD of all the     // array elements     $__gcd = 0;     for ($i = 0; $i < $n; $i++)      {         $__gcd = gcd($__gcd, $arr[$i]);     }          // Return the smallest prime factor     // of the gcd calculated     return smallestDivisor($__gcd); }      // Driver code $arr = array(2, 4, 8); $n = count($arr);  echo smallestInteger($arr, $n);  // This code is contributed by Srathore ?>

JavaScript

<script>     // Javascript implementation of the approach     function __gcd(a, b)      {          if (b == 0)              return a;          return __gcd(b, a % b);       }       // Function to find the smallest divisor      function smallestDivisor(x)      {          // if divisible by 2          if (x % 2 == 0)              return 2;           // iterate from 3 to sqrt(n)          for (let i = 3; i * i <= x; i += 2)          {              if (x % i == 0)                  return i;          }           return x;      }       // Function to return smallest possible integer     // which divides the whole array     function smallestInteger(arr, n)     {              // To store the GCD of all the array elements         let gcd = 0;         for (let i = 0; i < n; i++)             gcd = __gcd(gcd, arr[i]);          // Return the smallest prime factor         // of the gcd calculated         return smallestDivisor(gcd);     }          let arr = [ 2, 4, 8 ];     let n = arr.length;     document.write(smallestInteger(arr, n));  // This code is contributed by divyeshrabadiya07. </script>

Output

Time Complexity: O(n*(log(min(a, b))))
Auxiliary Space: O(1)

For multiple queries, we can precompute the smallest prime factors for numbers to a maximum value using a sieve.

C++

// C++ implementation of the approach #include <bits/stdc++.h> using namespace std;  const int MAX = 100005;  // To store the smallest prime factor int spf[MAX];  // Function to store spf of integers void sieve() {     memset(spf, 0, sizeof(spf));     spf[0] = 1;      // When gcd is 1 then the answer is -1     spf[1] = -1;     for (int i = 2; i * i < MAX; i++) {         if (spf[i] == 0) {             for (int j = i * 2; j < MAX; j += i) {                 if (spf[j] == 0) {                     spf[j] = i;                 }             }         }     }     for (int i = 2; i < MAX; i++) {         if (!spf[i])             spf[i] = i;     } }  // Function to return smallest possible integer // which divides the whole array int smallestInteger(int* arr, int n) {      // To store the GCD of all the array elements     int gcd = 0;     for (int i = 0; i < n; i++)         gcd = __gcd(gcd, arr[i]);      // Return the smallest prime factor     // of the gcd calculated     return spf[gcd]; }  // Driver code int main() {     sieve();     int arr[] = { 2, 4, 8 };     int n = sizeof(arr) / sizeof(arr[0]);     cout << smallestInteger(arr, n);      return 0; }

Java

// Java implementation of the approach class GFG  {  static int MAX = 100005;   // To store the smallest prime factor  static int spf[] = new int[MAX];   // Function to store spf of integers  static void sieve()  {      spf[0] = 1;       // When gcd is 1 then the answer is -1      spf[1] = -1;      for (int i = 2; i * i < MAX; i++)      {          if (spf[i] == 0)          {              for (int j = i * 2; j < MAX; j += i)             {                  if (spf[j] == 0)                  {                      spf[j] = i;                  }              }          }      }      for (int i = 2; i < MAX; i++)      {          if (spf[i] != 1)              spf[i] = i;      }  }   // Function to return smallest possible integer  // which divides the whole array  static int smallestInteger(int[] arr, int n)  {       // To store the GCD of all the array elements      int gcd = 0;      for (int i = 0; i < n; i++)          gcd = __gcd(gcd, arr[i]);       // Return the smallest prime factor      // of the gcd calculated      return spf[gcd];  }  static int __gcd(int a, int b)  {      if (b == 0)          return a;      return __gcd(b, a % b);       }  // Driver code  public static void main(String[] args)  {     sieve();      int arr[] = { 2, 4, 8 };      int n = arr.length;      System.out.println(smallestInteger(arr, n));  } }  /* This code contributed by PrinciRaj1992 */

Python3

# Python3 implementation of the approach MAX = 10005;  # To store the smallest prime factor spf = [0] * MAX;  # Function to store spf of integers def sieve():      spf[0] = 1;      # When gcd is 1 then the answer is -1     spf[1] = -1;     i = 2;     while (i * i < MAX):         if (spf[i] == 0):             for j in range(i * 2, MAX, i):                  if (spf[j] == 0):                     spf[j] = i;         i += 1;          for i in range(2, MAX):         if (spf[i] == 0):             spf[i] = i;  # find gcd of two no def __gcd(a, b):      if (b == 0):          return a;      return __gcd(b, a % b);   # Function to return smallest possible integer # which divides the whole array def smallestInteger(arr, n):          # To store the GCD of all the array elements     gcd = 0;     for i in range(n):         gcd = __gcd(gcd, arr[i]);      # Return the smallest prime factor     # of the gcd calculated     return spf[gcd];  # Driver code sieve(); arr = [ 2, 4, 8 ]; n = len(arr); print(smallestInteger(arr, n));  # This code is contributed by mits

// C# implementation of above approach  using System;      class GFG  {  static int MAX = 100005;   // To store the smallest prime factor  static int []spf = new int[MAX];   // Function to store spf of integers  static void sieve()  {      spf[0] = 1;       // When gcd is 1 then the answer is -1      spf[1] = -1;      for (int i = 2; i * i < MAX; i++)      {          if (spf[i] == 0)          {              for (int j = i * 2; j < MAX; j += i)             {                  if (spf[j] == 0)                  {                      spf[j] = i;                  }              }          }      }      for (int i = 2; i < MAX; i++)      {          if (spf[i] != 1)              spf[i] = i;      }  }   // Function to return smallest possible integer  // which divides the whole array  static int smallestInteger(int[] arr, int n)  {       // To store the GCD of all the array elements      int gcd = 0;      for (int i = 0; i < n; i++)          gcd = __gcd(gcd, arr[i]);       // Return the smallest prime factor      // of the gcd calculated      return spf[gcd];  }  static int __gcd(int a, int b)  {      if (b == 0)          return a;      return __gcd(b, a % b);       }  // Driver code  public static void Main(String[] args)  {     sieve();      int []arr = { 2, 4, 8 };      int n = arr.Length;      Console.WriteLine(smallestInteger(arr, n));  } }  // This code has been contributed by 29AjayKumar

PHP

<?php // PHP implementation of the approach  $MAX = 10005;  // To store the smallest prime factor $spf = array_fill(0, $MAX, 0);  // Function to store spf of integers function sieve() {     global $spf, $MAX;     $spf[0] = 1;      // When gcd is 1 then the answer is -1     $spf[1] = -1;     for ($i = 2; $i * $i < $MAX; $i++)     {         if ($spf[$i] == 0)          {             for ($j = $i * 2; $j < $MAX; $j += $i)              {                 if ($spf[$j] == 0)                  {                     $spf[$j] = $i;                 }             }         }     }     for ($i = 2; $i < $MAX; $i++)      {         if (!$spf[$i])             $spf[$i] = $i;     } }  // find gcd of two no function __gcd($a, $b)  {      if ($b == 0)          return $a;      return __gcd($b, $a % $b);       }   // Function to return smallest possible integer // which divides the whole array function smallestInteger($arr, $n) {     global $spf, $MAX;          // To store the GCD of all the array elements     $gcd = 0;     for ($i = 0; $i < $n; $i++)         $gcd = __gcd($gcd, $arr[$i]);      // Return the smallest prime factor     // of the gcd calculated     return $spf[$gcd]; }  // Driver code sieve(); $arr = array( 2, 4, 8 ); $n = count($arr); echo smallestInteger($arr, $n);  // This code is contributed by mits ?>

JavaScript

<script>     // Javascript implementation of above approach          let MAX = 100005;       // To store the smallest prime factor     let spf = new Array(MAX);      // Function to store spf of integers     function sieve()     {         spf[0] = 1;          // When gcd is 1 then the answer is -1         spf[1] = -1;         for (let i = 2; i * i < MAX; i++)         {             if (spf[i] == 0)             {                 for (let j = i * 2; j < MAX; j += i)                 {                     if (spf[j] == 0)                     {                         spf[j] = i;                     }                 }             }         }         for (let i = 2; i < MAX; i++)         {             if (spf[i] != 1)                 spf[i] = i;         }     }      // Function to return smallest possible integer     // which divides the whole array     function smallestInteger(arr, n)     {          // To store the GCD of all the array elements         let gcd = 0;         for (let i = 0; i < n; i++)             gcd = __gcd(gcd, arr[i]);          // Return the smallest prime factor         // of the gcd calculated         return spf[gcd];     }      function __gcd(a, b)     {         if (b == 0)             return a;         return __gcd(b, a % b);         }          sieve();     let arr = [ 2, 4, 8 ];     let n = arr.length;     document.write(smallestInteger(arr, n));  </script>

Output

How to Find the Smallest Number in an Array in C++?

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