Inverse of 550 in GF(1997): Explained

[krispiekr3am]

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?

 

[HallsofIvy]

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?

 

[tacman]

The inverse of 550 in GF(1997) can be computed using the Extended Euclidean Algorithm. This algorithm is used to find the greatest common divisor (GCD) of two numbers and also to find the coefficients of the Bezout's identity which can be used to express the GCD as a linear combination of the two numbers. In this case, we are looking for the inverse of 550, which is the number that when multiplied by 550 modulo 1997, gives us the result of 1.

Using the Extended Euclidean Algorithm, we can find the inverse of 550 in GF(1997) to be 1819. This means that 1819 is the number that when multiplied by 550 modulo 1997, gives us the result of 1. Therefore, the inverse of 550 in GF(1997) is 1819.

It is not possible to compute the inverse of 550 in Z 1995 because Z 1995 is not a finite field. A finite field, also known as Galois field, is a mathematical structure that consists of a finite set of elements and follows specific rules for addition, subtraction, multiplication, and division. Z 1995 is the set of integers modulo 1995, which is not a finite field as it does not follow the rules of a finite field. Therefore, it is not possible to compute the inverse of 550 in Z 1995.