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146x = 12 (mod 421)

I found out that 421 is a prime number and I still did not know how to start after going over the notes

I feel like it is similar to solve 7x = 1 (mod 180) if I know how to solve one of them, any tips? Thank you.

Bob
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Hints:

Do you know how to use the Euclidean Algorithm to find a positive integer $r$ such that $(146)r \equiv 1\pmod{421}?$ If so, given that $(146)x \equiv 12 \pmod{421},$ what could you say about $(146)(r)(x)$, which must be congruent to $(12)r \pmod{421}?$

user2661923
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  • Yes, I figured it out already. It was in my notes. So I just multiply both sides by the inverse right? – Bob Mar 09 '21 at 03:36
  • @Bob Yes, exactly. Also, I would encourage you to review [this article](https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question), so that going forward, your mathSE queries will be of high quality. – user2661923 Mar 09 '21 at 03:38
  • Thanks, I have another random question: since I now know how to solve modular equation, what about simplifying 4^103 mod 180 – Bob Mar 09 '21 at 03:39
  • @Bob The short answer is: find a Number Theory book that (at some point in the book) attacks that question, open the book to page 1, and go forward. I am not being deliberately obtuse. The question that you have asked requires some **moderately** heavy Number Theory machinery to master. It would be somewhat onerous for me to have to encapsulate this machinery in a series of comments. – user2661923 Mar 09 '21 at 03:44
  • Got it, but I think I have figured it out by myself again XD. Thanks for the hints and advice of mathSE. Have a good day :D – Bob Mar 09 '21 at 03:58
  • @Bob: regarding your other question, see [this](https://math.stackexchange.com/questions/81228/how-do-i-compute-ab-bmod-c-by-hand) – J. W. Tanner Mar 09 '21 at 04:24
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    @J.W.Tanner Nice response; good shooting the gap. – user2661923 Mar 09 '21 at 04:26
  • Please strive not to add more dupe answers to dupes of FAQs. – Bill Dubuque Mar 09 '21 at 09:05
  • @BillDubuque Good point. I've never considered that before. – user2661923 Mar 09 '21 at 10:06