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# Modular Division

• Difficulty Level : Medium
• Last Updated : 27 May, 2022

Given three positive numbers a, b and m. Compute a/b under modulo m. The task is basically to find a number c such that (b * c) % m = a % m.
Examples:

```Input  : a  = 8, b = 4, m = 5
Output : 2

Input  : a  = 8, b = 3, m = 5
Output : 1
Note that (1*3)%5 is same as 8%5

Input  : a  = 11, b = 4, m = 5
Output : 4
Note that (4*4)%5 is same as 11%5```

Following articles are prerequisites for this.
Modular multiplicative inverse
Extended Euclidean algorithms
Can we always do modular division?
The answer is “NO”. First of all, like ordinary arithmetic, division by 0 is not defined. For example, 4/0 is not allowed. In modular arithmetic, not only 4/0 is not allowed, but 4/12 under modulo 6 is also not allowed. The reason is, 12 is congruent to 0 when modulus is 6.
When is modular division defined?
Modular division is defined when modular inverse of the divisor exists. The inverse of an integer ‘x’ is another integer ‘y’ such that (x*y) % m = 1 where m is the modulus.
When does inverse exist? As discussed here, inverse a number ‘a’ exists under modulo ‘m’ if ‘a’ and ‘m’ are co-prime, i.e., GCD of them is 1.
How to find modular division?

```The task is to compute a/b under modulo m.
1) First check if inverse of b under modulo m exists or not.
a) If inverse doesn't exists (GCD of b and m is not 1),
print "Division not defined"
b) Else return  "(inverse * a) % m" ```

## C++

 `// C++ program to do modular division` `#include` `using` `namespace` `std;`   `// C++ function for extended Euclidean Algorithm` `int` `gcdExtended(``int` `a, ``int` `b, ``int` `*x, ``int` `*y);`   `// Function to find modulo inverse of b. It returns` `// -1 when inverse doesn't` `int` `modInverse(``int` `b, ``int` `m)` `{` `    ``int` `x, y; ``// used in extended GCD algorithm` `    ``int` `g = gcdExtended(b, m, &x, &y);`   `    ``// Return -1 if b and m are not co-prime` `    ``if` `(g != 1)` `        ``return` `-1;`   `    ``// m is added to handle negative x` `    ``return` `(x%m + m) % m;` `}`   `// Function to compute a/b under modulo m` `void` `modDivide(``int` `a, ``int` `b, ``int` `m)` `{` `    ``a = a % m;` `    ``int` `inv = modInverse(b, m);` `    ``if` `(inv == -1)` `       ``cout << ``"Division not defined"``;` `    ``else` `       ``cout << ``"Result of division is "` `<< (inv * a) % m;` `}`   `// C function for extended Euclidean Algorithm (used to` `// find modular inverse.` `int` `gcdExtended(``int` `a, ``int` `b, ``int` `*x, ``int` `*y)` `{` `    ``// Base Case` `    ``if` `(a == 0)` `    ``{` `        ``*x = 0, *y = 1;` `        ``return` `b;` `    ``}`   `    ``int` `x1, y1; ``// To store results of recursive call` `    ``int` `gcd = gcdExtended(b%a, a, &x1, &y1);`   `    ``// Update x and y using results of recursive` `    ``// call` `    ``*x = y1 - (b/a) * x1;` `    ``*y = x1;`   `    ``return` `gcd;` `}`   `// Driver Program` `int` `main()` `{` `    ``int`  `a  = 8, b = 3, m = 5;` `    ``modDivide(a, b, m);` `    ``return` `0;` `}`   `//this code is contributed by khushboogoyal499`

## C

 `// C program to do modular division` `#include `   `// C function for extended Euclidean Algorithm` `int` `gcdExtended(``int` `a, ``int` `b, ``int` `*x, ``int` `*y);`   `// Function to find modulo inverse of b. It returns` `// -1 when inverse doesn't` `int` `modInverse(``int` `b, ``int` `m)` `{` `    ``int` `x, y; ``// used in extended GCD algorithm` `    ``int` `g = gcdExtended(b, m, &x, &y);`   `    ``// Return -1 if b and m are not co-prime` `    ``if` `(g != 1)` `        ``return` `-1;`   `    ``// m is added to handle negative x` `    ``return` `(x%m + m) % m;` `}`   `// Function to compute a/b under modulo m` `void` `modDivide(``int` `a, ``int` `b, ``int` `m)` `{` `    ``a = a % m;` `    ``int` `inv = modInverse(b, m);` `    ``if` `(inv == -1)` `     ``printf` `(``"Division not defined"``);` `    ``else` `    ``{` `      ``int` `c = (inv * a) % m ;` `       ``printf` `(``"Result of division is %d"``, c) ;` `    ``}` `}`   `// C function for extended Euclidean Algorithm (used to` `// find modular inverse.` `int` `gcdExtended(``int` `a, ``int` `b, ``int` `*x, ``int` `*y)` `{` `    ``// Base Case` `    ``if` `(a == 0)` `    ``{` `        ``*x = 0, *y = 1;` `        ``return` `b;` `    ``}`   `    ``int` `x1, y1; ``// To store results of recursive call` `    ``int` `gcd = gcdExtended(b%a, a, &x1, &y1);`   `    ``// Update x and y using results of recursive` `    ``// call` `    ``*x = y1 - (b/a) * x1;` `    ``*y = x1;`   `    ``return` `gcd;` `}`   `// Driver Program` `int` `main()` `{` `    ``int`  `a  = 8, b = 3, m = 5;` `    ``modDivide(a, b, m);` `    ``return` `0;` `}`

## Java

 `// java program to do modular division`   `import` `java.io.*;` `import` `java.lang.Math;`   `public` `class` `GFG {`   `    ``static` `int` `gcd(``int` `a,``int` `b){` `        ``if` `(b == ``0``){` `            ``return` `a;` `        ``}         ` `        ``return` `gcd(b, a % b);` `    ``}` `    `  `    ``// Function to find modulo inverse of b. It returns ` `    ``// -1 when inverse doesn't ` `    ``// modInverse works for prime m` `    ``static` `int` `modInverse(``int` `b,``int` `m){` `        ``int` `g = gcd(b, m) ;` `        ``if` `(g != ``1``)` `            ``return` `-``1``;` `        ``else` `        ``{` `            ``//If b and m are relatively prime, ` `            ``//then modulo inverse is b^(m-2) mode m ` `            ``return` `(``int``)Math.pow(b, m - ``2``) % m;` `        ``}` `    ``}` `    `  `    ``// Function to compute a/b under modulo m ` `    ``static` `void` `modDivide(``int` `a,``int` `b,``int` `m){` `        ``a = a % m;` `        ``int` `inv = modInverse(b,m);` `        ``if``(inv == -``1``){` `            ``System.out.println(``"Division not defined"``);` `        ``}   ` `        ``else``{` `             ``System.out.println(``"Result of Division is "` `+ ((inv*a) % m));` `        ``}` `    ``}  ` `    `  `    ``// Driver Program ` `    ``public` `static` `void` `main(String[] args) {` `        ``int` `a = ``8``;` `        ``int` `b = ``3``;` `        ``int` `m = ``5``;` `        ``modDivide(a, b, m);` `    ``}` `}`   `// The code is contributed by Gautam goel (gautamgoel962)`

## Python3

 `# Python3 program to do modular division` `import` `math`   `# Function to find modulo inverse of b. It returns ` `# -1 when inverse doesn't ` `# modInverse works for prime m` `def` `modInverse(b,m):` `    ``g ``=` `math.gcd(b, m) ` `    ``if` `(g !``=` `1``):` `        ``# print("Inverse doesn't exist") ` `        ``return` `-``1` `    ``else``: ` `        ``# If b and m are relatively prime, ` `        ``# then modulo inverse is b^(m-2) mode m ` `        ``return` `pow``(b, m ``-` `2``, m)`     `# Function to compute a/b under modulo m ` `def` `modDivide(a,b,m):` `    ``a ``=` `a ``%` `m` `    ``inv ``=` `modInverse(b,m)` `    ``if``(inv ``=``=` `-``1``):` `        ``print``(``"Division not defined"``)` `    ``else``:` `        ``print``(``"Result of Division is "``,(inv``*``a) ``%` `m)`   `# Driver Program ` `a ``=` `8` `b ``=` `3` `m ``=` `5` `modDivide(a, b, m)`   `# This code is Contributed by HarendraSingh22`

## C#

 `using` `System;`   `// C# program to do modular division` `class` `GFG {`   `  ``// Recursive Function  to find` `  ``// GCD of two numbers` `  ``static` `int` `gcd(``int` `a,``int` `b){` `    ``if` `(b == 0){` `      ``return` `a;` `    ``}         ` `    ``return` `gcd(b, a % b);` `  ``}`   `  ``// Function to find modulo inverse of b. It returns ` `  ``// -1 when inverse doesn't ` `  ``// modInverse works for prime m` `  ``static` `int` `modInverse(``int` `b,``int` `m){` `    ``int` `g = gcd(b, m) ;` `    ``if` `(g != 1){` `      ``return` `-1;` `    ``}        ` `    ``else` `    ``{`   `      ``//If b and m are relatively prime, ` `      ``//then modulo inverse is b^(m-2) mode m ` `      ``return` `(``int``)Math.Pow(b, m - 2) % m;` `    ``}` `  ``}`   `  ``// Function to compute a/b under modulo m ` `  ``static` `void` `modDivide(``int` `a,``int` `b,``int` `m){` `    ``a = a % m;` `    ``int` `inv = modInverse(b,m);` `    ``if``(inv == -1){` `      ``Console.WriteLine(``"Division not defined"``);` `    ``}   ` `    ``else``{` `      ``Console.WriteLine(``"Result of Division is "` `+ ((inv*a) % m));` `    ``}` `  ``}  `   `  ``// Driver Code ` `  ``static` `void` `Main() {` `    ``int` `a = 8;` `    ``int` `b = 3;` `    ``int` `m = 5;` `    ``modDivide(a, b, m);` `  ``}` `}`   `// The code is contributed by Gautam goel (gautamgoel962)`

## Javascript

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## PHP

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Output:

`Result of division is 1`

Modular division is different from addition, subtraction and multiplication.
One difference is division doesn’t always exist (as discussed above). Following is another difference.

```Below equations are valid
(a * b) % m = ((a % m) * (b % m)) % m
(a + b) % m = ((a % m) + (b % m)) % m

// m is added to handle negative numbers
(a - b + m) % m = ((a % m) - (b % m) + m) % m

But,
(a / b) % m may NOT be same as ((a % m)/(b % m)) % m

For example, a = 10, b = 5, m = 5.
(a / b) % m is 2, but ((a % m) / (b % m)) % m
is not defined.```

References:
http://www.doc.ic.ac.uk/~mrh/330tutor/ch03.html