Problem in finding the inverse element in modular arithmetic

Question

I'm new to modular arithmetic but it still is something that I think is pure mathematics that I don't get.

I have this problem 3 * d ≡ 1 mod 13. So I need to find the d which is the inverse of 3. I did find a solution that I understand:

3 * d ≡ 1 mod 13

Which can be written

3^(13-2) % 13 = 9 or pow(3, 13-2, 13) = 9

So that I understand, and this is the accepted answer. But my thought was a bit different:

My thought to calculate d was to divide both sides by 3 which gives us d = (1/3) * 1 mod 13

And that in python can be written pow(3, -1, 13) or pow(3, -1) % 13 which also gives 9 as a result.

But at first, I thought that it was wrong. How this calculation can give us 9? I mean 3^-1 = 1/3. And 1/3 mod 13 = 1/3.

In which part of my thinking am I wrong?

Answer 1

Notation: for integer $m>0$, we write $u\equiv v\pmod m$ when $u-v$ is a multiple of $m$. We write $u=v\bmod m$ when additionally $0\le u<m$. In the former, $\pmod m$ qualifies the congruence sign(s) $\equiv$ on its left. In the later, $\bmod$ is an operator.

The modular modular inverse $x$ of integer $a$ modulo integer $m>1$ is defined when $\gcd(a,m)=1$. It then is the uniquely defined integer $x$ such that $0<x<m$ and $a\,x\bmod m=1$. It's noted $x=a^{-1}\bmod n$. It holds $3^{-1}\bmod13=9$ because $\gcd(3,13)=1$, and $0<9<13$, and $3\times9=27$, and $27-1$ is a multiple of $13$.

A general method to determine if $a^{-1}\bmod n$ is defined, and compute it, is to find $x$, $y$, $g$ for the Bézout identity $a\,x+m\,y=g$. The inverse $a^{-1}\bmod n$ is defined when the Greatest Common Divisor $g=1$, and then it holds $a\,x\equiv1\pmod m$, and therefore $a^{-1}\bmod n\,=\,x\bmod n$.

One method to compute $x$, $y$, $g$ for the Bézout identity $a\,x+m\,y=g$ is the extended Euclidean algorithm. When computing the modular inverse, we do not need $y$, which allows to maintain two less variables. This slight variant deals only with non-negative quantities.

In modern Python, when operator pow has three integer parameters, and the second (the exponent) is a negative integer, and the third is greater than one, it's used a method equivalent to the above to determine if the multiplicative inverse exists; in the affirmative it's computed and raised to the absolute value of the exponent modulo the modulus. That's why pow(3, -1, 13) yields 9. That's the generic method to compute the multiplicative inverse in modern Python.

When $m$ is prime, $\gcd(x,m)=1$ is the same as $x\bmod m\ne0$. And when that holds, Fermat Little Theorem tells that $x^{m-1}\bmod m=1$. It follows that then, the modular modular inverse $y$ of $x$ is $x^{m-2}\bmod m$. This is why pow(3, 13-2, 13) also yields the multiplicative inverse of $3$ modulo $13$.

_{Note: pow(3, -1) % 13 asks Python to compute $\frac13$, that is $0.3333\ldots$, then reduce it modulo $13$, which leaves it unchanged.}

_{Note: Python defines pow(u,e,1) to be 0, and handles negative modulus so that pow(u,e,m) is -pow(u,e,-m).}