Proof Consider the map II(x) aX Suppose x*y and ax=ay mod m then a(x-y)=0 mod m. So if gcd(a, m)=1 then x=y mod m. Therefore, it is a bijection. Therefore, every ax=b mod m as a unique solution In particular ax=I mod m has a solution which implies that a has an inverseProof • Consider the map: Pa (x) = ax. Suppose x  y and ax = ay mod m then a(x-y) = 0 mod m. So if gcd(a,m) =1, then x = y mod m. Therefore, it is a bijection. Therefore, every ax = b mod m has a unique solution • In particular ax = 1 mod m has a solution, which implies that a has an inverse