Let and be distinct primes, and let satisfy . Then for any , .
For some reason, even in published works, this important cryptographic result is often only half proved. Here is a full proof.
Proof: We consider two cases: (i) is relatively prime to both and (and hence relatively prime to ), and (ii) is a multiple of one of the primes, say . We note that can’t be a product of both primes, for then it would be greater than their product, thus violating the statement of the theorem.
(i) By definition, for some integer . Since , then by Euler’s theorem
since both sides are equal to . We can also write , and since is relatively prime to , then by Euler’s theorem again and so
By the Chinese remainder theorem, (1) and (2) together prove as required.