So i have a problem like this
34x ≡ 77 (mod 89)
This is how i try to solve it but it doesn't seem to work :( If anyone can give me a hint on how to proceed from what i have down below would mean a lot thanks.
I first use the Extended Euclidean algorithm to find the inverse down below
Since the gcd = 1 i look at the the fact that
37*77-(-32)*89=1
Therefore going from there since we live under mod 89 we can cancel out (-32)*89 so i continue with multiplying the inverse to my original equation.
37*34x ≡ 37*77 (mod 89)
Shouldn't it come out to be: x ≡ 2849 (mod 89) and let that be the answer? I know i am going wrong somewhere all help is appreciated.
Also sorry for formatting if it is bad quite new at this.
I think you've got it backwards. Given your equation, you have
$$37(77)\equiv1\pmod{89}$$
In other words, you have that $37$ is the inverse of $77$. What you want is the inverse of $34$. Multiplying both sides by that will yield $x$ on the left hand side.