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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, 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 – 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