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# Sum of main diagonal elements in a Matrix which are prime

Given a matrix mat[][] of R rows and C columns. The task is to find the sum of all elements from the main diagonal which are prime numbers
Note: The main diagonals are the ones that occur from Top Left of Matrix Down To Bottom Right Corner.

Examples:

Input: R = 3, C = 3, mat[][] = {{1, 2, 3}, {0, 1, 2}, {0, 4, 2}}
Output:
Explanation:
Elements from main diagonal are { 1, 1, 2}, out of these only ‘2’ is a prime number.
Therefore, the sum of diagonal elements which are prime = 2.

Input: R = 4, C = 4, mat[][] = { {1, 2, 3, 4}, { 0, 7, 21, 12}, { 1, 2, 3, 6}, { 3, 5, 2, 31}}
Output: 41
Explanation:
Elements from main diagonal are { 1, 7, 3, 31}, out of these {7, 3, 31} are prime numbers.
Therefore, the sum of diagonal elements which are prime = 7 + 3 + 31 = 41.

Naive Approach:

1. Traverse the given matrix and check if the current element belongs to main diagonal or not.
2. If element belongs to the main diagonal and it is a prime number then add value of the element to totalSum.
3. After, traversal of the matrix print the value of totalSum.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function checks whether a number``// is prime or not``bool` `isPrime(``int` `n)``{``    ``if` `(n < 2) {``        ``return` `false``;``    ``}``    ``// Iterate to check primality of n``    ``for` `(``int` `i = 2; i < n; i++) {``        ``if` `(n % i == 0)``            ``return` `false``;``    ``}` `    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``void` `primeDiagonalElementSum(``    ``int``* mat,``    ``int` `r, ``int` `c)``{` `    ``// Initialise total sum as 0``    ``int` `totalSum = 0;` `    ``// Iterate the given matrix mat[][]``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``for` `(``int` `j = 0; j < c; j++) {` `            ``int` `temp = *((mat + i * c) + j);` `            ``// If element belongs to main``            ``// diagonal and is prime``            ``if` `((i == j) && isPrime(temp))``                ``totalSum += (temp);``        ``}``    ``}` `    ``// Print the total sum``    ``cout << totalSum << endl;``}` `// Driver Code``int` `main()``{``    ``int` `R = 4, C = 5;` `    ``// Given Matrix``    ``int` `mat[4][5] = { { 1, 2, 3, 4, 2 },``                      ``{ 0, 3, 2, 3, 9 },``                      ``{ 0, 4, 1, 2, 8 },``                      ``{ 1, 2, 3, 6, 6 } };` `    ``// Function Call``    ``primeDiagonalElementSum((``int``*)mat, R, C);``    ``return` `0;``}`

## Java

 `// Java program for the above approach``class` `GFG{` `// Function checks whether a number``// is prime or not``static` `boolean` `isPrime(``int` `n)``{``    ``if` `(n < ``2``)``    ``{``        ``return` `false``;``    ``}` `    ``// Iterate to check primarility of n``    ``for``(``int` `i = ``2``; i < n; i++)``    ``{``       ``if` `(n % i == ``0``)``           ``return` `false``;``    ``}` `    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``static` `void` `primeDiagonalElementSum(``int` `[][]mat,``                                    ``int` `r, ``int` `c)``{``    ` `    ``// Initialise total sum as 0``    ``int` `totalSum = ``0``;` `    ``// Iterate the given matrix mat[][]``    ``for``(``int` `i = ``0``; i < r; i++)``    ``{``       ``for``(``int` `j = ``0``; j < c; j++)``       ``{``          ``int` `temp = mat[i][j];``          ` `          ``// If element belongs to main``          ``// diagonal and is prime``          ``if` `((i == j) && isPrime(temp))``              ``totalSum += (temp);``       ``}``    ``}` `    ``// Print the total sum``    ``System.out.print(totalSum + ``"\n"``);``}` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `R = ``4``, C = ``5``;` `    ``// Given Matrix``    ``int` `mat[][] = { { ``1``, ``2``, ``3``, ``4``, ``2` `},``                    ``{ ``0``, ``3``, ``2``, ``3``, ``9` `},``                    ``{ ``0``, ``4``, ``1``, ``2``, ``8` `},``                    ``{ ``1``, ``2``, ``3``, ``6``, ``6` `} };` `    ``// Function Call``    ``primeDiagonalElementSum(mat, R, C);``}``}` `// This code is contributed by gauravrajput1`

## Python3

 `# Python3 program for the above approach` `# Function checks whether a number``# is prime or not``def` `isPrime(n):` `    ``if` `(n < ``2``):``        ``return` `False`` ` `    ``# Iterate to check primarility of n``    ``for` `i ``in` `range``(``2``, n):``        ``if` `(n ``%` `i ``=``=` `0``):``            ``return` `False`` ` `    ``return` `True` `# Function calculates the sum of``# prime elements of main diagonal``def` `primeDiagonalElementSum(mat, r, c):`` ` `    ``# Initialise total sum as 0``    ``totalSum ``=` `0``;`` ` `    ``# Iterate the given matrix mat[][]``    ``for` `i ``in` `range``(r):``        ``for` `j ``in` `range``(c):``            ``temp ``=` `mat[i][j]`` ` `            ``# If element belongs to main``            ``# diagonal and is prime``            ``if` `((i ``=``=` `j) ``and` `isPrime(temp)):``                ``totalSum ``+``=` `(temp)`` ` `    ``# Print the total sum``    ``print``(totalSum)` `# Driver code``if` `__name__``=``=``"__main__"``:``    ` `    ``R ``=` `4``    ``C ``=` `5`` ` `    ``# Given Matrix``    ``mat ``=` `[ [ ``1``, ``2``, ``3``, ``4``, ``2` `],``            ``[ ``0``, ``3``, ``2``, ``3``, ``9` `],``            ``[ ``0``, ``4``, ``1``, ``2``, ``8` `],``            ``[ ``1``, ``2``, ``3``, ``6``, ``6` `] ]`` ` `    ``# Function call``    ``primeDiagonalElementSum(mat, R, C)``    ` `# This code is contributed by rutvik_56`

## C#

 `// C# program for the above approach``using` `System;` `class` `GFG{` `// Function checks whether a number``// is prime or not``public` `static` `bool` `isPrime(``int` `n)``{``    ``if` `(n < 2)``    ``{``        ``return` `false``;``    ``}` `    ``// Iterate to check primarility of n``    ``for``(``int` `i = 2; i < n; i++)``    ``{``       ``if` `(n % i == 0)``           ``return` `false``;``    ``}``    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``public` `static` `void` `primeDiagonalElementSum(``int` `[,]mat,``                                           ``int` `r, ``int` `c)``{``    ` `    ``// Initialise total sum as 0``    ``int` `totalSum = 0;` `    ``// Iterate the given matrix mat[][]``    ``for``(``int` `i = 0; i < r; i++)``    ``{``       ``for``(``int` `j = 0; j < c; j++)``       ``{``          ``int` `temp = mat[i, j];``          ` `          ``// If element belongs to main``          ``// diagonal and is prime``          ``if` `((i == j) && isPrime(temp))``              ``totalSum += (temp);``       ``}``    ``}` `    ``// Print the total sum``    ``Console.WriteLine(totalSum);``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `R = 4, C = 5;` `    ``// Given Matrix``    ``int` `[,]mat = { { 1, 2, 3, 4, 2 },``                   ``{ 0, 3, 2, 3, 9 },``                   ``{ 0, 4, 1, 2, 8 },``                   ``{ 1, 2, 3, 6, 6 } };` `    ``// Function Call``    ``primeDiagonalElementSum(mat, R, C);``}``}` `// This code is contributed by SoumikMondal`

## Javascript

 ``

Output:

`3`

Time Complexity: O(R*C*K), where K is the maximum element in the matrix.
Auxiliary Space: O(1)

Efficient Approach: We can optimize the naive approach by optimizing the primality test of the number. Below are the steps for optimizing the primality test:

1. Instead of checking till N, we can check till sqrt(N) as the larger factor of N must be a multiple of smaller factor that has been already checked.
2. The algorithm can be improved further by observing that all primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = -1, 0, 1, 2, 3, or 4.
3. As 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So a more efficient method is to test if N is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function checks whether a number``// is prime or not``bool` `isPrime(``int` `n)``{``    ``// Corner cases``    ``if` `(n <= 1)``        ``return` `false``;``    ``if` `(n <= 3)``        ``return` `true``;` `    ``// This is checked so that we can skip``    ``// middle five numbers in below loop``    ``if` `(n % 2 == 0 || n % 3 == 0)``        ``return` `false``;` `    ``for` `(``int` `i = 5; i * i <= n; i = i + 6)``        ``if` `(n % i == 0 || n % (i + 2) == 0)``            ``return` `false``;` `    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``void` `primeDiagonalElementSum(``    ``int``* mat,``    ``int` `r, ``int` `c)``{` `    ``// Initialise total sum as 0``    ``int` `totalSum = 0;` `    ``// Iterate the given matrix mat[][]``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``for` `(``int` `j = 0; j < c; j++) {` `            ``int` `temp = *((mat + i * c) + j);` `            ``// If element belongs to main``            ``// diagonal and is prime``            ``if` `((i == j) && isPrime(temp))``                ``totalSum += (temp);``        ``}``    ``}` `    ``// Print the total sum``    ``cout << totalSum << endl;``}` `// Driver Code``int` `main()``{``    ``int` `R = 4, C = 5;` `    ``// Given Matrix``    ``int` `mat[4][5] = { { 1, 2, 3, 4, 2 },``                      ``{ 0, 3, 2, 3, 9 },``                      ``{ 0, 4, 1, 2, 8 },``                      ``{ 1, 2, 3, 6, 6 } };` `    ``// Function Call``    ``primeDiagonalElementSum((``int``*)mat, R, C);``    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;` `class` `GFG{` `// Function checks whether a number``// is prime or not``static` `boolean` `isPrime(``int` `n)``{``    ` `    ``// Corner cases``    ``if` `(n <= ``1``)``        ``return` `false``;``    ``if` `(n <= ``3``)``        ``return` `true``;` `    ``// This is checked so that we can skip``    ``// middle five numbers in below loop``    ``if` `(n % ``2` `== ``0` `|| n % ``3` `== ``0``)``        ``return` `false``;` `    ``for``(``int` `i = ``5``; i * i <= n; i = i + ``6``)``        ``if` `(n % i == ``0` `|| n % (i + ``2``) == ``0``)``            ``return` `false``;` `    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``static` `void` `primeDiagonalElementSum(``int``[][] mat,``                                    ``int` `r, ``int` `c)``{` `    ``// Initialise total sum as 0``    ``int` `totalSum = ``0``;` `    ``// Iterate the given matrix mat[][]``    ``for``(``int` `i = ``0``; i < r; i++)``    ``{``        ``for``(``int` `j = ``0``; j < c; j++)``        ``{``            ``int` `temp = mat[i][j];` `            ``// If element belongs to main``            ``// diagonal and is prime``            ``if` `((i == j) && isPrime(temp))``                ``totalSum += (temp);``        ``}``    ``}` `    ``// Print the total sum``    ``System.out.print(totalSum + ``"\n"``);``}` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `R = ``4``, C = ``5``;` `    ``// Given Matrix``    ``int` `mat[][] = { { ``1``, ``2``, ``3``, ``4``, ``2` `},``                    ``{ ``0``, ``3``, ``2``, ``3``, ``9` `},``                    ``{ ``0``, ``4``, ``1``, ``2``, ``8` `},``                    ``{ ``1``, ``2``, ``3``, ``6``, ``6` `} };` `    ``// Function call``    ``primeDiagonalElementSum(mat, R, C);``}``}` `// This code is contributed by Rajput-Ji`

## Python3

 `# Python3 program for the above approach` `# Function checks whether a number``# is prime or not``def` `isPrime(n):` `    ``# Corner cases``    ``if` `(n <``=` `1``):``        ``return` `False``    ``if` `(n <``=` `3``):``        ``return` `True` `    ``# This is checked so that we can skip``    ``# middle five numbers in below loop``    ``if` `(n ``%` `2` `=``=` `0` `or` `n ``%` `3` `=``=` `0``):``        ``return` `False``        ` `    ``i ``=` `5``    ``while` `i ``*` `i <``=` `n:``        ``if` `(n ``%` `i ``=``=` `0` `or` `n ``%` `(i ``+` `2``) ``=``=` `0``):``            ``return` `False``            ` `        ``i ``+``=` `6``        ` `    ``return` `True` `# Function calculates the sum of``# prime elements of main diagonal``def` `primeDiagonalElementSum(mat, r, c):` `    ``# Initialise total sum as 0``    ``totalSum ``=` `0` `    ``# Iterate the given matrix mat[][]``    ``for` `i ``in` `range``(r):``        ``for` `j ``in` `range``(c):``            ``temp ``=` `mat[i][j]``            ` `            ``# If element belongs to main``            ``# diagonal and is prime``            ``if` `((i ``=``=` `j) ``and` `isPrime(temp)):``                ``totalSum ``+``=` `(temp)` `    ``# Print the total sum``    ``print``(totalSum)``    ` `# Driver Code``R, C ``=` `4``, ``5` `# Given Matrix``mat ``=` `[ [ ``1``, ``2``, ``3``, ``4``, ``2` `],``        ``[ ``0``, ``3``, ``2``, ``3``, ``9` `],``        ``[ ``0``, ``4``, ``1``, ``2``, ``8` `],``        ``[ ``1``, ``2``, ``3``, ``6``, ``6` `] ]` `# Function Call``primeDiagonalElementSum(mat, R, C)` `# This code is contributed by divyeshrabadiya07`

## C#

 `// C# program for the above approach``using` `System;``class` `GFG{` `// Function checks whether a number``// is prime or not``static` `bool` `isPrime(``int` `n)``{``    ` `    ``// Corner cases``    ``if` `(n <= 1)``        ``return` `false``;``    ``if` `(n <= 3)``        ``return` `true``;` `    ``// This is checked so that we can skip``    ``// middle five numbers in below loop``    ``if` `(n % 2 == 0 || n % 3 == 0)``        ``return` `false``;` `    ``for``(``int` `i = 5; i * i <= n; i = i + 6)``        ``if` `(n % i == 0 || n % (i + 2) == 0)``            ``return` `false``;` `    ``return` `true``;``}` `// Function calculates the sum of``// prime elements of main diagonal``static` `void` `primeDiagonalElementSum(``int``[,] mat,``                                    ``int` `r, ``int` `c)``{` `    ``// Initialise total sum as 0``    ``int` `totalSum = 0;` `    ``// Iterate the given matrix [,]mat``    ``for``(``int` `i = 0; i < r; i++)``    ``{``        ``for``(``int` `j = 0; j < c; j++)``        ``{``            ``int` `temp = mat[i,j];` `            ``// If element belongs to main``            ``// diagonal and is prime``            ``if` `((i == j) && isPrime(temp))``                ``totalSum += (temp);``        ``}``    ``}` `    ``// Print the total sum``    ``Console.Write(totalSum + ``"\n"``);``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `R = 4, C = 5;` `    ``// Given Matrix``    ``int` `[,]mat = { { 1, 2, 3, 4, 2 },``                    ``{ 0, 3, 2, 3, 9 },``                    ``{ 0, 4, 1, 2, 8 },``                    ``{ 1, 2, 3, 6, 6 } };` `    ``// Function call``    ``primeDiagonalElementSum(mat, R, C);``}``}` `// This code is contributed by Rohit_ranjan`

## Javascript

 ``

Output:

`3`

Time Complexity: O(R*C*sqrt(K)), where K is the maximum element in the matrix.
Auxiliary Space: O(1)

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