# Remainder with 7 for large numbers

Given a large number as a string, find the remainder of number when divided by 7.

Examples :

```Input : num = 1234
Output : 2

Input : num = 1232
Output : 0

Input : num = 12345
Output : 4
```

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Simple Approach is to convert a string into number and perform the mod operation. But this approach will not work for long strings.

There exists an approach that works for large numbers also. Below are the steps.

Let the given number be “num”

1. We use a series 1, 3, 2, -1, -3, -2 to find the remainder (Intuition behind the series is discussed below).
2. Initialize the result as 0.
3. Traverse num from end and above series from beginning. For every traversed digit, multiply it with next digit of series, and add the multiplication to result.
4. Keep repeating step 3 while there are more digits to process. If there are more than 6(number of terms in series) digits, then start traversing the series from beginning.
5. After each step, we keep doing result = result % 7 to make sure that result remains less than 7.

Illustration :

```let us take above Example where number is 12345.

We reverse the number from end and series from
the beginning and keep adding multiplication to
the result.
(12345 % 7) = (5*1 + 4*3 + 3*2 + 2*(-1) + 1*(-3)) % 7
= (5 + 12 + 6 - 2 - 3)%7
= (18) % 7
= 4
hence 4 will be the remainder when we divide
the number 12345 by 7. ```

How does this series formula work?
Below is the intuition behind the series

```1  % 7 = 1
10 % 7 = 3
100 % 7 = 2
1000 % 7 = 6 = (-1) % 7
10000 % 7 = 4 = (-3) % 7
100000 % 7 = 5 = (-2) % 7

The series repeats itself for larger powers
1000000 % 7 = 1
10000000 % 7 = 3
..............
...............

The above property of modular division with 7 and
associative properties of modular arithmetic are
the basis of the approach used here.
```

Implementation:

## C++

 `// C++ program to find remainder of a large ` `// number when divided by 7. ` `#include ` `using` `namespace` `std; ` ` `  `/* Function which returns Remainder after dividing ` `   ``the number by 7 */` `int` `remainderWith7(string num) ` `{ ` `    ``// This series is used to find remainder with 7 ` `    ``int` `series[] = {1, 3, 2, -1, -3, -2}; ` ` `  `    ``// Index of next element in series ` `    ``int` `series_index = 0; ` ` `  `    ``int` `result = 0;  ``// Initialize result ` ` `  `    ``// Traverse num from end ` `    ``for` `(``int` `i=num.size()-1; i>=0; i--) ` `    ``{ ` `        ``/* Find current digit of nun */` `        ``int` `digit = num[i] - ``'0'``; ` ` `  `        ``// Add next term to result ` `        ``result += digit * series[series_index]; ` ` `  `        ``// Move to next term in series ` `        ``series_index = (series_index + 1) % 6; ` ` `  `        ``// Make sure that result never goes beyond 7. ` `        ``result %= 7; ` `    ``} ` ` `  `    ``// Make sure that remainder is positive ` `    ``if` `(result < 0) ` `      ``result = (result + 7) % 7; ` ` `  `    ``return` `result; ` `} ` ` `  `/* Driver program */` `int` `main() ` `{ ` `    ``string str = ``"12345"``; ` `    ``cout << ``"Remainder with 7 is "` `         ``<< remainderWith7(str); ` `    ``return` `0; ` `} `

## Java

 `// Java program to find remainder of a large ` `// number when divided by 7. ` ` `  `class` `GFG ` `{ ` `    ``// Function which return Remainder  ` `    ``// after dividingthe number by 7  ` `    ``static` `int` `remainderWith7(String num) ` `    ``{ ` `        ``// This series is used to  ` `        ``// find remainder with 7 ` `        ``int` `series[] = {``1``, ``3``, ``2``, -``1``, -``3``, -``2``}; ` `     `  `        ``// Index of next element in series ` `        ``int` `series_index = ``0``; ` `     `  `        ``// Initialize result ` `        ``int` `result = ``0``;  ` `     `  `        ``// Traverse num from end ` `        ``for` `(``int` `i = num.length() - ``1``; i >= ``0``; i--) ` `        ``{ ` `            ``/* Find current digit of nun */` `            ``int` `digit = num.charAt(i) - ``'0'``; ` `     `  `            ``// Add next term to result ` `            ``result += digit * series[series_index]; ` `     `  `            ``// Move to next term in series ` `            ``series_index = (series_index + ``1``) % ``6``; ` `     `  `            ``// Make sure that result never goes beyond 7. ` `            ``result %= ``7``; ` `        ``} ` `     `  `        ``// Make sure that remainder is positive ` `        ``if` `(result < ``0``) ` `        ``result = (result + ``7``) % ``7``; ` `     `  `        ``return` `result; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `main (String[] args) ` `    ``{ ` `        ``String str = ``"12345"``; ` `        ``System.out.print(``"Remainder with 7 is "` `                          ``+remainderWith7(str)); ` `    ``} ` `} ` ` `  `// This code is contributed by Anant Agarwal. `

## Python3

 `# Python3 program to find remainder of ` `# a large number when divided by 7. ` ` `  `# Function which return Remainder ` `# after dividing the number by 7  ` `def` `remainderWith7(num): ` `     `  `    ``# This series is used to ` `    ``# find remainder with 7 ` `    ``series ``=` `[``1``, ``3``, ``2``, ``-``1``, ``-``3``, ``-``2``]; ` `     `  `    ``# Index of next element ` `    ``# in series ` `    ``series_index ``=` `0``; ` `     `  `    ``# Initialize result ` `    ``result ``=` `0``; ` `     `  `    ``# Traverse num from end ` `    ``for` `i ``in` `range``((``len``(num) ``-` `1``), ``-``1``, ``-``1``): ` `         `  `        ``# Find current digit of num ` `        ``digit ``=` `ord``(num[i]) ``-` `48``; ` `         `  `        ``# Add next term to result ` `        ``result ``+``=` `digit ``*` `series[series_index]; ` `         `  `        ``# Move to next term in series ` `        ``series_index ``=` `(series_index ``+` `1``) ``%` `6``; ` `         `  `        ``# Make sure that result ` `        ``# never goes beyond 7. ` `        ``result ``%``=` `7``; ` `     `  `    ``# Make sure that remainder ` `    ``# is positive ` `     `  `    ``if` `(result < ``0``): ` `        ``result ``=` `(result ``+` `7``) ``%` `7``; ` `    ``return` `result; ` ` `  `# Driver Code ` `str` `=` `"12345"``; ` `print``(``"Remainder with 7 is"``, ` `       ``remainderWith7(``str``)); ` ` `  `# This code is contributed by mits `

## C#

 `// C# program to find the remainder of  ` `// a large number when divided by 7. ` `using` `System; ` ` `  `class` `GFG ` `{ ` `    ``// Function which return Remainder  ` `    ``// after dividingthe number by 7  ` `    ``static` `int` `remainderWith7(String num) ` `    ``{ ` `        ``// This series is used to  ` `        ``// find remainder with 7 ` `        ``int` `[]series = {1, 3, 2, -1, -3, -2}; ` `     `  `        ``// Index of next element in series ` `        ``int` `series_index = 0; ` `     `  `        ``// Initialize result ` `        ``int` `result = 0;  ` `     `  `        ``// Traverse num from end ` `        ``for` `(``int` `i = num.Length - 1; i >= 0; i--) ` `        ``{ ` `            ``/* Find current digit of nun */` `            ``int` `digit = num[i] - ``'0'``; ` `     `  `            ``// Add next term to result ` `            ``result += digit * series[series_index]; ` `     `  `            ``// Move to next term in series ` `            ``series_index = (series_index + 1) % 6; ` `     `  `            ``// Make sure that result never goes beyond 7. ` `            ``result %= 7; ` `        ``} ` `     `  `        ``// Make sure that remainder is positive ` `        ``if` `(result < 0) ` `        ``result = (result + 7) % 7; ` `     `  `        ``return` `result; ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main () ` `    ``{ ` `        ``String str = ``"12345"``; ` `        ``Console.Write(``"Remainder with 7 is "` `+  ` `                         ``remainderWith7(str)); ` `    ``} ` `} ` ` `  `// This code is contributed by nitin mittal. `

## PHP

 `= 0; ``\$i``--) ` `    ``{ ` `         `  `        ``// Find current digit of num  ` `        ``\$digit` `= ``\$num``[``\$i``] - ``'0'``; ` ` `  `        ``// Add next term to result ` `        ``\$result` `+= ``\$digit` `* ` `                   ``\$series``[``\$series_index``]; ` ` `  `        ``// Move to next term in series ` `        ``\$series_index` `= (``\$series_index` `+ 1) % 6; ` ` `  `        ``// Make sure that result ` `        ``// never goes beyond 7. ` `        ``\$result` `%= 7; ` `    ``} ` ` `  `    ``// Make sure that remainder  ` `    ``// is positive ` `    ``if` `(``\$result` `< 0) ` `    ``\$result` `= (``\$result` `+ 7) % 7; ` ` `  `    ``return` `\$result``; ` `} ` ` `  `// Driver Code ` `{ ` `    ``\$str` `= ``"12345"``; ` `    ``echo` `"Remainder with 7 is "``, ` `         ``(remainderWith7(``\$str``)); ` `    ``return` `0; ` `} ` ` `  `// This code is contributed by nitin mittal. ` `?> `

Output:

```Remainder with 7 is 4
```

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