Finding congruent equation whilst using modulo arithmetic

Question

My question is

Notice that if r x + s y = t then r x ≡ t mod s . By using Euclid’s algorithm to find integers x and y satisfying
4647 x + 1145 y = 1 , find solutions of the following congruence's.
(a) 1145 a ≡ 1 mod 4647
(b) 4647 b ≡ 1 mod 1145
(c) 1145 c ≡ 127 mod 4647

If i can figure out part (a) i believe i can get the rest reasonably easily i'm so close its painful.

so here is where I'm at,

Therefore: 1145a ≡ 1 mod 4647 is

4647(x) + 1145(y) = 1 Going by Euclid's method

Then solved x and y and got:

X = 188 and Y = - 763

Therefor 4647(188) + 1145(-763) = 1 (which is true)

So now I've solved for X and Y but now i need to work out a to solve the congruence, how do i go about this?

Thank you for your time

Answer 1

So you've obtained that $4647(188) + 1145(-763) = 1$ now consider this congruence modulo $4647$. We get that $1145(-763) \equiv 1 mod 4647$. Do you see it now?