Inverse of 550 in GF(1997): Explained

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compute the inverse of 550 in GF (1997). Notice GF (p), Zp, or Ip I covered before consisting of all integers 0, 1, .., p -1 modulo p are the same thing with different names. Can we compute the inverse of 550 in Z 1995 ? Why?
 
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Looks to me like a direct quote from a textbook! The (multiplicative) inverse of 550 (mod 1997) is an integer x< 1997 such that 550x= 1 (mod 1997) or such that 550x= 1997m+ 1 for some integer m. Do you know how to use Euclid's division algorithm (repeated division) to solve the Diophantine equation 550x- 1997m= 1?

As far as the inverse of 550 mod Z1995 is concerned, I notice that 550 and 1995 are both divisible by 5. Do you know why that is important?
 

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