最小整数> 1，该整数除以给定数组的每个元素

给定数组arr [] ，任务是找到最小的整数(除1以外)，该整数对给定数组的每个元素进行除法。

例子：

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.

方法：我们知道整个数组的GCD将是将数组的每个元素相除的最大整数。如果GCD = 1，则不可能将整个数组相除。但是，如果GCD> 1，则存在一个整数，该整数将数组完全分割。例如，

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.

下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
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#

// 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

输出：

2

对于多个查询，我们可以使用筛子预先计算出最小的质数因子，直到达到最大值。

C++

// C++ implementation of the approach
#include 
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＃

// C# implemenatation 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

输出：

2

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