# Product of two integers a and b (less then integer n) is congruent to 1 if a or b integer is relativity prime to n .

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## Theorem:-

      If n is any positive integer then for every positive integer m<n and relatively prime to n then ∃ positive integer b<n :
m*b≡1(mod n).


## Proof:-

      Let n is any positive integer and m is positive integer less then n and 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.