Question

Chinese Remainder Theorem

Answer 1

The theorem states that:
Suppose \[\gcd(m,n)\] = 1
given \[a, b\]
there should be exactly one solution to the system of congurences:
\[x = a (\mod m); x = b (\mod n)\]

Now that any solution to any congruences should have infinite number solutions. So what does the theorem exactly mean?

Thanks

Answer 2

If \(x_0\) is one solution, then an integer \(x\) satisfies the congruences if and only if \(x\) is of the form \(x = x_0 +kM\) for some integer \(k\), and \(M=mn\).

Answer 3

so the theorem actually states any 2 solution x_i, x_j must satisfy x_i = x_j (mod mn)

Answer 4

yes, that's what I remember.