multiplicative inverse in modular arithmetic


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I am a beginner in modular arithmetic area, and i am studying modular arithmetic. i came across a question, if i have mod 5 or mod 7, since they are smaller numbers it is easier to find their multiplicative inverse as they are prime, but anything that is even for example , i couldnt understand the following practice problem:

106x (is congruent to) 10 mod 284

As far as i understand the multiplicative inverse exists if gcd(106,284)=1, since 284 is not prime, since

gcd(106, 284 mod 106) = gcd(106,72)= gcd(72,106)= gcd(34,72)=gcd(4,34)=gcd(2,4)=gcd(0,2) hence gcd of (106,284)=2.

1) but how do i know for sure that the multiplicative inverse doesnot exist for this qs. Can i somehow prove it.

2) what if 284 is replaced by 283?

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