Chinese Remainder Theorem: How Can It Help Solve Modulo Residue Problems?

[saadsarfraz]

Chinese remainder theorem, urgent!

Homework Statement

This is an attempt to make the Chinese Remainder Theorem more concrete.
Let m = 206 and n = 125. You may use the fact that 89n - 54m = 1.

(a) What does the Chinese Remainder Theorem have to say about pairs
of residues modulo 206 and 125? How do you know that it applies?

(b) Find residues x and y modulo mn = 25,750 where
x = 1 (mod m); x = 0 (mod n);
y = 0 (mod m); y = 1 (mod n):

(c) Find a residue z modulo 25,750 where
z = 10 (mod m); z = 24 (mod n):

Homework Equations

see above

The Attempt at a Solution

for part a) CRT says that there is a unique N modulo mn such that N=a mod m and N=b mod n. I don't know what "How do you know that it applies means". The problem set is due in a couple of hours. I would be very grateful for any help.

 

[saadsarfraz]

I think for part b) x=413=1 mod 206 and y=412=0 mod 206 makes sense. but i don't know if x and y have to be the same for the two mods m and n.