elementary number theory - How to solve this congruence $17x equiv 1 pmod{23}$?

Given $17x \equiv 1 \pmod{23}$

How to solve this linear congruence? All hints are welcome.

edit: I know the Euclidean Algorithm and know how to solve the equation $17m+23n=1$ but I don't know how to compute x with the use of m or n.

Answer

To do modular division I do this:

an - bm = c where c is dividend, b is modulo and a is divisor, then n is quotient

17n - 23m = 1

Then using euclidean algorithm, reduce to gcd(a,b) and record each calculation

As described by http://mathworld.wolfram.com/DiophantineEquation.html

17 23 $\quad$ 14 19

17 6 $\quad\;\;$ 14 5

11 6 $\quad\;\;\;\;$ 9 5

5 6 $\quad\;\;\;\;\;$ 4 5

5 1 $\quad\;\;\;\;\;$ 4 1

1 1 $\quad\;\;\;\;\;$ 0 1

Left column is euclidean algorithm, Right column is reverse procedure

Therefore $17*19 - 23*14 = 1$, i.e. n=19 and m=14.

The result is that 1/17 ≡ 19 mod 23

this method might not be as quick as the other posts, but this is what I have implemented in code. The others could also be, but I thought I would share my method.