User:Bilal Raza

product of two integers m and b congruent to 1 at modulo prime n

File:Theorem on congruence

Theorem:-

       If n is any prime then for every positive integer m<n their exist positive integer b<n :
                                  'm*b≡1(mod n).

Proof:-

       Let n is any prime and m is positive integer less then n.
       we known all integer less then any prime n number are co-prime to n
       that's way we can able to write
                                  (m,n)=1       where 1 is greatest common divisor of m and n
       we also known from number theory that greatest common divisor of ant two integers can be
       written as combination of that numbers,
       therefore their exist s,t belong to integers :
                                 1=ms+tn
             this implies        1-ms=tn
             this implies        n|(1-ms)
             this implies        ms≡1(mod n)
       now if s<n then we have done.
       But if s>n then we can take b≡s(mod n) which will give the same result
             so                  bm≡1(mod n)
                         hence prove.