Given an integer N( 2 <= N <= 10^9 ), split the number into one or more parts(possibly none), where each part must be greater than 1. The task is to find the minimum possible sum of the second largest divisor of all the splitting numbers.
Examples:
Input : N = 27 Output : 3 Explanation : Split the given number into 19, 5, 3. Second largest divisor of each number is 1. So, sum is 3. Input : N = 19 Output : 1 Explanation : Don't make any splits. Second largest divisor of 19 is 1. So, sum is 1
Approach:
The idea is based on Goldbach’s conjecture.
- When the number is prime, then the answer will be 1.
- When a number is even then it can always be expressed as a sum of 2 primes. So, the answer will be 2.
- When the number is odd,
- When N-2 is prime, then the number can be express as the sum of 2 primes, that are 2 and N-2, then the answer will be 2.
- Otherwise, the answer will always be 3.
Below is the implementation of the above approach:
C++
// CPP program to find sum of all second largest divisor // after splitting a number into one or more parts #include <bits/stdc++.h> using namespace std;
// Function to find a number is prime or not bool prime( int n)
{ if (n == 1)
return false ;
// If there is any divisor
for ( int i = 2; i * i <= n; ++i)
if (n % i == 0)
return false ;
return true ;
} // Function to find the sum of all second largest divisor // after splitting a number into one or more parts int Min_Sum( int n)
{ // If number is prime
if (prime(n))
return 1;
// If n is even
if (n % 2 == 0)
return 2;
// If the number is odd
else {
// If N-2 is prime
if (prime(n - 2))
return 2;
// There exists 3 primes x1, x2, x3
// such that x1 + x2 + x3 = n
else
return 3;
}
} // Driver code int main()
{ int n = 27;
// Function call
cout << Min_Sum(n);
return 0;
} |
Java
// Java program to Sum of all second largest // divisors after splitting a number into one or more parts import java.io.*;
class GFG {
// Function to find a number is prime or not static boolean prime( int n)
{ if (n == 1 )
return false ;
// If there is any divisor
for ( int i = 2 ; i * i <= n; ++i)
if (n % i == 0 )
return false ;
return true ;
} // Function to find the sum of all second largest divisor // after splitting a number into one or more parts static int Min_Sum( int n)
{ // If number is prime
if (prime(n))
return 1 ;
// If n is even
if (n % 2 == 0 )
return 2 ;
// If the number is odd
else {
// If N-2 is prime
if (prime(n - 2 ))
return 2 ;
// There exists 3 primes x1, x2, x3
// such that x1 + x2 + x3 = n
else
return 3 ;
}
} // Driver code public static void main (String[] args) {
int n = 27 ;
// Function call
System.out.println( Min_Sum(n));
}
} // This code is contributed by anuj_6 |
Python3
# Python 3 program to find sum of all second largest divisor # after splitting a number into one or more parts from math import sqrt
# Function to find a number is prime or not def prime(n):
if (n = = 1 ):
return False
# If there is any divisor
for i in range ( 2 , int (sqrt(n)) + 1 , 1 ):
if (n % i = = 0 ):
return False
return True
# Function to find the sum of all second largest divisor # after splitting a number into one or more parts def Min_Sum(n):
# If number is prime
if (prime(n)):
return 1
# If n is even
if (n % 2 = = 0 ):
return 2
# If the number is odd
else :
# If N-2 is prime
if (prime(n - 2 )):
return 2
# There exists 3 primes x1, x2, x3
# such that x1 + x2 + x3 = n
else :
return 3
# Driver code if __name__ = = '__main__' :
n = 27
# Function call
print (Min_Sum(n))
# This code is contributed by # Surendra_Gangwar |
C#
// C# program to Sum of all second largest // divisors after splitting a number into one or more parts using System;
class GFG
{ // Function to find a number is prime or not static bool prime( int n)
{ if (n == 1)
return false ;
// If there is any divisor
for ( int i = 2; i * i <= n; ++i)
if (n % i == 0)
return false ;
return true ;
} // Function to find the sum of all second largest divisor // after splitting a number into one or more parts static int Min_Sum( int n)
{ // If number is prime
if (prime(n))
return 1;
// If n is even
if (n % 2 == 0)
return 2;
// If the number is odd
else {
// If N-2 is prime
if (prime(n - 2))
return 2;
// There exists 3 primes x1, x2, x3
// such that x1 + x2 + x3 = n
else
return 3;
}
} // Driver code public static void Main ()
{ int n = 27;
// Function call
Console.WriteLine( Min_Sum(n));
} } // This code is contributed by anuj_6 |
Javascript
<script> // Javascript program to find sum of all second largest divisor // after splitting a number into one or more parts // Function to find a number is prime or not function prime(n)
{ if (n == 1)
return false ;
// If there is any divisor
for (let i = 2; i * i <= n; ++i)
if (n % i == 0)
return false ;
return true ;
} // Function to find the sum of all second largest divisor // after splitting a number into one or more parts function Min_Sum(n)
{ // If number is prime
if (prime(n))
return 1;
// If n is even
if (n % 2 == 0)
return 2;
// If the number is odd
else {
// If N-2 is prime
if (prime(n - 2))
return 2;
// There exists 3 primes x1, x2, x3
// such that x1 + x2 + x3 = n
else
return 3;
}
} // Driver code let n = 27;
// Function call
document.write(Min_Sum(n));
// This code is contributed by Mayank Tyagi </script> |
Output:
3
Time complexity: O(sqrt(N))
Auxiliary Space: O(1)