Question

Can someone help me explain the proof for the uniqueness of the Chinese Remainder Theorem. I get most of the proof but I just don't get the end.

Answer 1

The uniqueness proof for the Chinese Remainder Theorem is based on the fact that if two pairs of remainders are equal modulo the respective pairwise coprime moduli, then the original numbers must also be equal modulo the product of the moduli.

Step-by-step explanation:

The uniqueness proof for the Chinese Remainder Theorem is based on the fact that if two pairs of remainders are equal modulo the respective pairwise coprime moduli, then the original numbers must also be equal modulo the product of the moduli.

Here's the step-by-step explanation:

1. Suppose we have two pairs of remainders: (a,b) and (c,d), both satisfying the Chinese Remainder Theorem conditions for the same set of pairwise coprime moduli.
2. Since the moduli are pairwise coprime, each pair of congruences can be solved separately.
3. We can write a = c + k*M and b = d + k*N, where M is the product of all moduli except the one corresponding to a, and N is the product of all moduli except the one corresponding to b.
4. From the second congruence, d ≡ b mod N. Since the moduli are pairwise coprime, N and M are also coprime, which means that d ≡ b mod M as well.
5. Combining the congruences, we have a ≡ c + k*M ≡ c mod M. Similarly, b ≡ d mod N ≡ d mod M.
6. Therefore, a ≡ c mod M and b ≡ d mod M, which implies that (a,b) ≡ (c,d) mod M.

Thus, the two pairs of remainders are equivalent modulo the product of the moduli, which proves the uniqueness of the Chinese Remainder Theorem.