How can I find inverse of something in modular arithmetic?

Question

I have this problem: $16x ≡ 22 \quad(\text{mod} 35)$

Now I think I have to find the inverse of that. How can I?

I thought it will be $\quad x_0 = 16^{(24-1)} *22 \quad (\text{mod} 35) \quad$ but I think it is not the answer. Can someone please help me how to find it and how to make it step by step?

Answer 1

$$x\equiv16^{-1}22\equiv2^{-4}22\pmod{35}$$

Now using Carmichael Function, $\lambda(35)=12,$ $$2^{-4}\equiv2^8\equiv11\pmod{35}$$

Alternatively,

As $16x\equiv1\pmod5\iff x\equiv1\pmod5$

and $16x\equiv1\pmod7\iff 2x\equiv1\iff x\equiv2^{-1}\equiv4\pmod7$

Now apply Chinese Remainder Theorem