calculate modular multiplicative inverse

Author: danrusei

README.md

@@ -83,7 +83,7 @@ WIP, the descriptions of the below `unsolved yet` problems can be found in the [
 ## [Number Theory](https://github.com/danrusei/algorithms_with_Go/tree/main/numbers)
 
 - [] Modular Exponentiation
-- [] Modular multiplicative inverse
+- [x] [Modular multiplicative inverse](https://github.com/danrusei/algorithms_with_Go/tree/main/numbers/multiplicative)
 - [] Primality Test | Set 2 (Fermat Method)
 - [] Euler’s Totient Function
 - [x] [Sieve of Eratosthenes](https://github.com/danrusei/algorithms_with_Go/tree/main/numbers/eratosthenes)

numbers/multiplicative/README.md

@@ -0,0 +1,25 @@
+# Modular multiplicative inverse 
+
+Source: [GeeksforGeeks](https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/amp/)  
+Source: [Wikipedia - Modular Multiplicative Inverse](https://en.wikipedia.org/wiki/Modular_multiplicative_inverse)
+
+Given two integers ‘a’ and ‘m’, find modular multiplicative inverse of ‘a’ under modulo ‘m’.  
+The modular multiplicative inverse is an integer ‘x’ such that.
+
+a x ≅ 1 (mod m)
+
+The value of x should be in {0, 1, 2, … m-1}, i.e., in the range of integer modulo m.  
+The multiplicative inverse of “a modulo m” exists if and only if a and m are relatively prime (i.e., if gcd(a, m) = 1).
+
+## Example
+
+> Input:  a = 3, m = 11  
+> Output: 4  
+> Since (4*3) mod 11 = 1, 4 is modulo inverse of 3(under 11).  
+> One might think, 15 also as a valid output as "(15*3) mod 11"
+is also 1, but 15 is not in ring {0, 1, 2, ... 10}, so not
+valid.
+>
+> Input:  a = 10, m = 17  
+> Output: 12  
+> Since (10*12) mod 17 = 1, 12 is modulo inverse of 10(under 17).

numbers/multiplicative/main.go

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+package main
+
+import "fmt"
+
+func modInverse(a int, m int) int {
+	m0 := m
+	x, y := 1, 0
+
+	if m == 1 {
+		return 0
+	}
+
+	for a > 1 {
+		// q is quotient
+		q := a / m
+		t := m
+
+		// m is remainder now, process same as Euclid's algo
+		m = a % m
+		a = t
+		t = y
+
+		// update y and x
+		y = x - q*y
+		x = t
+	}
+
+	// make x positive
+	if x < 0 {
+		x += m0
+	}
+
+	return x
+}
+
+func main() {
+	a, m := 3, 11
+	fmt.Printf("Modular multiplicative inverse is %d\n", modInverse(a, m))
+}