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?

https://math.stackexchange.com/questions/407478/solving-a-linear-congruence/407482 — lab bhattacharjee, May 05 '20 at 09:13
Literally the same question How can I calculate this congruence? 16x ≡ 22 (mod 35) — Cheong Sik Feng, May 05 '20 at 09:26
I added some dupe links which give many methods for computing modular inverses. — Bill Dubuque, May 05 '20 at 15:50
$\bmod 35!:,\ x\equiv \dfrac{22}{16}\equiv \dfrac{44}{32}\equiv \dfrac{9}{-3}\equiv -3,$ by Gauss's algorithm $\ \ $ — Bill Dubuque, May 05 '20 at 15:54

score 0 · Answer 1 · answered May 05 '20 at 09:18

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

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

How can I find inverse of something in modular arithmetic?

1 Answers1