Thread: systems of congruences

Hello. I am completely stumped by this problem:

x = a1 (mod m1)
x = a2 (mod m2)
...
x = an (mod mn)

Let mj be arbitrary positive integers. Show that there is a simultaneous solution of this system if and only if ai= aj (mod(mi, mj)) for all pairs of the indices i,j for which 1 <= i < j <= r

Thanks for the help.

What do you mean by "mod(mi,mj)"?

I mean the greatest common divisor of mi and mj as the modulo. mi is and mj is

This in the commonly known Chinese Remainder Theorem.
Use Google to get some explanations.
There are proofs available on the internet.