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I am attempting to solve $ 769x\equiv1066 \mod 2022 $

Using the Euclidean Algorithm I have found the following:

$ 823\cdot 769 - 313\cdot2022 = 1 $

I am attempting to represent the solution in the form $ x\equiv a\mod n $ where $ 0 \le a\le n $

Where do $823$, $1066$ and $-313$ fit into the answer to the solution...?

Ben
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  • See the proof in the linked dupe, i.e. reducing the the bezout equation mod $2022$ yields $$\bmod 2022!:\ 769(823)\equiv 1\underset{\times 1066}\Longrightarrow 769\underbrace{(823)1066}_{\textstyle x}\equiv 1066\qquad\qquad$$ – Bill Dubuque Jan 31 '22 at 13:07

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Solution is $x \equiv 1066\cdot 823 \mod 2022$, since $823$ is the inverse of $769$ modulo $2022$, i.e., $823\cdot 769\equiv 1 \mod 2022$.

Wuestenfux
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  • Please strive not to add more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jan 31 '22 at 13:08
  • Ah yes I understand. Not confident with rearranging congruence equations yet. Although $a$ is larger than $n$ here, what should I do to $a$ ...? – Ben Jan 31 '22 at 20:07