# Sum of Digits in a^n till a single digit

Given two numbers a and n, the task is to find the single sum of digits of a^n (pow(a, n)). In single digit sum, we keep doing sum of digit until a single digit is left.

Examples:

```Input : a = 5, n = 4
Output : 4
5^4 = 625 = 6+2+5 = 13
Since 13 has two digits, we
sum again 1 + 3 = 4.

Input : a = 2, n = 8
Output : 4
2^8=256 = 2+5+6 = 13 = 1+3 = 4```

A naive approach is to first find a^n, then find sum of digits in a^n using the approach discussed here.

The above approach may cause overflow. A better solution is based on below observation.

```int res = 1;
for (int i=1; i<=n; i++)
{
res = res*a;
res = digSum(res);
}

Here digSum() finds single digit sum
of res. Please refer this for details
of digSum().```

Illustration of above pseudo code:

For example, let a = 5, n = 4.
After first iteration,
res = 5
After second iteration,
res = 7 (Note : 2 + 5 = 7)
After third iteration,
res = 8 (Note : 3 + 5 = 8)
After 4th iteration,
res = 4 (Note : 4 + 0 = 4)

We can write a function similar to a fast modular exponentiation to evaluate digSum(a^n) which evaluates this in log(n) steps.
Below is the implementation of above approach:

## C++

 `// CPP program to find single digit``// sum of a^n.``#include ``using` `namespace` `std;` `// This function finds single digit``// sum of n.``int` `digSum(``int` `n) ``{ ``    ``if` `(n == 0) ``    ``return` `0; ``    ``return` `(n % 9 == 0) ? 9 : (n % 9); ``} ` `// Returns single digit sum of a^n.``// We use modular exponentiation technique.``int` `powerDigitSum(``int` `a, ``int` `n)``{``    ``int` `res = 1;``    ``while` `(n) {``        ``if` `(n % 2 == 1) {``            ``res = res * digSum(a);``            ``res = digSum(res);``        ``}``        ``a = digSum(digSum(a) * digSum(a));``        ``n /= 2;``    ``}` `    ``return` `res;``}` `// Driver code``int` `main()``{``    ``int` `a = 9, n = 4;``    ``cout << powerDigitSum(a, n);``    ``return` `0;``}`

## Java

 `// Java program to find single digit ``// sum of a^n. ` `import` `java.util.*;``import` `java.lang.*;``import` `java.io.*;` `class` `GFG{``    ` `    ` `// This function finds single digit ``// sum of n. ``static` `int` `digSum(``int` `n) ``{ ``    ``if` `(n == ``0``) ``    ``return` `0``; ``    ``return` `(n % ``9` `== ``0``) ? ``9` `: (n % ``9``); ``} ` `// Returns single digit sum of a^n. ``// We use modular exponentiation technique. ``static` `int` `powerDigitSum(``int` `a, ``int` `n) ``{ ``    ``int` `res = ``1``; ``    ``while` `(n>``0``) { ``        ``if` `(n % ``2` `== ``1``) { ``            ``res = res * digSum(a); ``            ``res = digSum(res); ``        ``} ``        ``a = digSum(digSum(a) * digSum(a)); ``        ``n /= ``2``; ``    ``} ` `    ``return` `res; ``} ` `// Driver code``public` `static` `void` `main(String args[]) ``{ ``    ``int` `a = ``9``, n = ``4``; ``    ``System.out.print(powerDigitSum(a, n)); ``}``} `

## Python 3

 `# Python 3 Program to find single digit ``# sum of a^n. ` `# This function finds single digit ``# sum of n.``def` `digSum(n) :` `    ``if` `n ``=``=` `0` `:``        ``return` `0` `    ``elif` `n ``%` `9` `=``=` `0` `:``        ``return` `9` `    ``else` `:``        ``return` `n ``%` `9` `# Returns single digit sum of a^n. ``# We use modular exponentiation technique.``def` `powerDigitSum(a, n) :` `    ``res ``=` `1``    ``while``(n) :` `        ``if` `n ``%``2` `=``=` `1` `:``            ``res ``=` `res ``*` `digSum(a)``            ``res ``=` `digSum(res)` `        ``a ``=` `digSum(digSum(a) ``*` `digSum(a))``        ``n ``/``/``=` `2` `    ``return` `res`  `# Driver Code``if` `__name__ ``=``=` `"__main__"` `:` `    ``a, n ``=` `9``, ``4``    ``print``(powerDigitSum(a, n))` `# This code is contributed by ANKITRAI1`

## C#

 `// C# program to find single ``// digit sum of a^n. ``class` `GFG``{` `// This function finds single ``// digit sum of n. ``static` `int` `digSum(``int` `n) ``{ ``    ``if` `(n == 0) ``    ``return` `0; ``    ``return` `(n % 9 == 0) ? ``                      ``9 : (n % 9); ``} ` `// Returns single digit sum of a^n. ``// We use modular exponentiation ``// technique. ``static` `int` `powerDigitSum(``int` `a, ``int` `n) ``{ ``    ``int` `res = 1; ``    ``while` `(n > 0) ``    ``{ ``        ``if` `(n % 2 == 1) ``        ``{ ``            ``res = res * digSum(a); ``            ``res = digSum(res); ``        ``} ``        ``a = digSum(digSum(a) * digSum(a)); ``        ``n /= 2; ``    ``} ` `    ``return` `res; ``} ` `// Driver code``static` `void` `Main() ``{ ``    ``int` `a = 9, n = 4; ``    ``System.Console.WriteLine(powerDigitSum(a, n)); ``}``} ` `// This Code is contributed by mits`

## PHP

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## Javascript

 ``

Output:
`9`

Time Complexity: O(log2n)// since in the powerDigitSum function in every call the value of n is divided by 2 until it reaches 1 thus the algorithm takes logarithmic time to execute.

Auxiliary Space: O(1) // since no extra array is used so the space taken by the algorithm is constant

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