calculus - Getting an X for Chinese Remainder Theorem (CRT)

how do I get modulo equations to satisfy a given X in CRT.

For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of Mignotte's threshold secret sharing scheme. More precisely given in my example k = n = 4, and the product of any k - 1 is smaller then X how come simply computing the remainder of each won't give equations that solve to X = 1234.

In the case of the example,

x = 4 mod 5
x = 2 mod 7
x = 2 mod 11
x = 12 mod 13

Which resolves to 31264 (won't CRT produce the smallest?)

Any hints?

Answer

The final result of the CRT calculation must be reduced modulo 5 x 7 x 11 x 13 = 5005. This gives the correct answer.