Finding a solution to $144x \equiv 4 \pmod{233}$.

Question

I want to find the solution to this problem.

Find a solution to $144x \equiv 4 \pmod{233}$.

Can someone guide me on how to do it?

Answer 1

First you determine if 233 is prime or not, in this case it is so we can find an an inverse of 144 modulo 233. By Euclid's algorithm we have $233(-55)+144(89)$=$1$ in which case we have $144(89)$=$\equiv 1 \pmod{233}$ which means that $144^{-1}$$\equiv 89 \pmod{233}$. So multiplying the original congruence through by 89 we have $(144)(89)x\equiv\ x\equiv 4(89) \pmod{233}\equiv 356 \pmod{233}$. So $x\equiv 356\pmod{233}\equiv 123 \pmod{233}$.