# Understanding Wikipedia claim: modular multiplicative inverse of $17\pmod{\phi(3233)}$ is $2753$

I have been reading the wiki article about Modular multiplicative inverse and I don't understand it. Can you explain it to me in better way.

To be more specific I am trying to understand the RSA algorithm and I stuck in their working example paragraph 5. I don't understand the math behind this calculation:

The modular multiplicative inverse of $17\pmod{\phi(3233)}$ yielding $2753$.

• Which part don't you understand? Do you see that the given number is in fact the inverse? – Tobias Kildetoft Feb 25 '14 at 10:30
• Check this answer math.stackexchange.com/a/437978/72031 It uses different numbers to explain RSA. You can see that the inverse is calculated using the Euclid's algorithm for HCF/GCD – Paramanand Singh Feb 25 '14 at 10:30

## 1 Answer

What they refer to as a "worked example" is just a verification of the fact that the multiplicative inverse of $17$ modulo $\phi(3233) = 3120$ is $2753$.

In order to find the multiplicative inverse of $17$ modulo $3120$, suppose $17x \equiv 1 \bmod 3120$. Then $3120 \mid (17x - 1)$ so $17x - 1 = 3120y$ for some $y \in \mathbb{Z}$. Rearranging, we obtain the equation $17x - 3120y = 1$. Now we use the Euclidean algorithm to find the greatest common divisor of $17$ and $3120$:

\begin{align*} 3120 &= 183\times 17 + 9\\ 17 &= 1\times 9 + 8\\ 9 &= 1\times 8 + 1. \end{align*}

So the greatest common divisor of $17$ and $3120$ is $1$; note, if it weren't $1$, then the equation $17x - 3120y = 1$ would have no solutions. Using back substitution, we have

\begin{align*} 1 &= 9 - 1\times 8\\ &= 9 - 1\times(17 - 1\times 9)\\ &= 2\times 9 - 1\times 17\\ &= 2\times(3120 - 183\times 17) - 1\times 17\\ &= 2\times 3120 - 367\times 17. \end{align*}

Therefore $x = -367$, $y = 2$ is a solution to $17x - 3120y = 1$. So $(-367)\times 17 \equiv 1 \bmod 3120$. Note that for any $t \in \mathbb{Z}$, $(-367 + 3120t)\times 17 \equiv -367\times 17 \bmod 3120 \equiv 1\bmod 3120$. Taking $t = 1$, we see that $-367 + 3120\times 1 = 2753$.