9.5. Multiplicative Inverse Modulo N

Now, I define operation multiplication modulo N that is denoted by this notation ( . ) N. This notation is for multiplication modulo N, that means you multiply a and b and then take the remainder. That is our definition. Now, I say an integer b to be the multiplicative inverse of another integer a modulo N if a times b modulo N gives me 1, that means, if you multiply a with b and then take the modulo N and if the value is 1, then I will say that value b is the inverse of a (multiplicative inverse).

We start by defining what it means to multiply two integers under a specified modulus, N. The operation is called multiplication modulo N, and it's written as a . b mod N, indicating the process of multiplying a and b and then taking the remainder after division by N. Now, introducing the concept of a multiplicative inverse, we say that an integer b is the multiplicative inverse of a modulo N if the product a * b gives a remainder of 1 when divided by N. Therefore, finding a multiplicative inverse means finding a b that satisfies this condition.

Think of the multiplicative inverse like figuring out how many candies you need to divide evenly among friends to ensure nobody gets more than one extra candy. For instance, if you have 5 candies and want to share them among 3 friends, you'd find how many candies each one gets and what remains. If 2 candies are left after attempting to share, you would adjust until each has enough candies to keep things fair, meaning you 'invert' the imbalance created by the initial division.

Now, another interesting fact here is the following: if b is the multiplicative inverse of a, then any number of the form b ± kN, ∀ k ∈ ℤ, will also be an inverse of a.

This part states a key property of the multiplicative inverse. If b is the multiplicative inverse of a modulo N, then you can generate an infinite number of inverses by adding or subtracting multiples of N from b. This is because, according to modular arithmetic rules, when you adjust b by a multiple of N, the result is still congruent to the original relation (ab mod N = 1). So if we take b + kN or b - kN, the equation still holds true.

Imagine you have a specific way to score points in a game, which directly corresponds to finding the right moves (the 'inverse') that give you a winning edge. If a particular move (b) turns out to be a winning strategy, then creating variations of that move by adding or subtracting the rounds of the game (like multiples of N) might also work – this gives players multiple strategies (inverse options) while still bringing them to victory.

But now, the interesting question is when can we say that the multiplicative inverse modulo N for a number a exists - is it the case that for every a for every modulus N I can say that the multiplicative inverse modular and exist? Unfortunately, the answer is no, there is only under certain conditions I can say that multiplicative inverse modular and exist.

Not every number has a multiplicative inverse under every modulus. The existence of a multiplicative inverse is conditional. Specifically, the multiplicative inverse of a exists if and only if a is coprime to N, meaning the greatest common divisor (GCD) of a and N must be 1. This is essential for defining the inverse in modular arithmetic.

Consider a scenario where you can only trade items with a specific kind of currency. If your items cannot be exchanged (because they are too common or not of equal value), then you can't find the right pairs to swap for what you need. This is similar to how only specific pairs of numbers (those whose GCD is 1) can interact to yield a useful result, like a multiplicative inverse in the realm of numbers.

Let us first prove the sufficiency condition using Bezout’s theorem. That means, assume that you have a number a and a modulus N such that they are co-prime. If that is the case then I have to show that I can find out the multiplicative inverse of a using Bezout’s theorem and using extended Euclid algorithm.

The sufficiency condition states that if a is coprime to N (i.e., GCD(a, N) = 1), then there exist integers s and t such that sa + tN = 1. By applying the extended Euclid's algorithm, we can find these coefficients, deriving the multiplicative inverse of a modulo N from the equation. It shows that we can indeed constructively find the inverse when a and N are coprime.

Think about navigating a situation or environment where two paths (the number a and the modulus N) diverge but share a common goal (the GCD being 1). If you find routes that complement each other effectively, you create a successful path to where you need to be - analogous to finding the specific integers that yield the desired outcome in number theory.

Now, I want to prove the necessity condition. Namely, I want to show that if at all the multiplicative inverse of a exists then it implies that the number a is co-prime to your modulus.

This proof establishes that if a has a multiplicative inverse, then a and N cannot share any common factors apart from 1, hence, a and N must be coprime. The reasoning is rooted in the definition of the multiplicative inverse - if ab mod N = 1, then both components must not share factors that could prevent the product from yielding a remainder of 1 upon division by N.

Imagine having a unique combination lock. If you can unlock (invert) it (where your number a has an inverse), it suggests that your combination (a) couldn't use any shared digits (factors) with the lock's mechanism (N). If they did, you'd simply get stuck (no remainder of 1). Thus the uniqueness ensures the lock can only work when they are coprime.

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