Thread: Chinese Remainder Theorem

I understand how to do the Chinese Remainder Theorem when all of the moduli are pairwise relatively prime, but I don't understand how to do it when all of the moduli are NOT pairwise relatively prime.

For example, say we have
x≡9 (mod 12)
x≡3 (mod 9)
x≡7 (mod 10)

According to my teacher this becomes
x≡1 (mod 4)
x≡3 (mod 9)
x≡2 (mod 5).
I don't understand how this was arrived at though. Could someone please clearly explain the process?

x≡9 (mod 12)
x≡3 (mod 9)
x≡7 (mod 10)

x≡1 (mod 4)
x≡3 (mod 9)
x≡2 (mod 5)

Uhm, Chinese Remainder Theorem just ensure you there exist an x such that all those conditions are satisfied.

Well, you know that x≡9 (mod 12), then and (both are true).
x≡1 (mod 4).

x≡3 (mod 9) Remained unchanged.

x≡7 (mod 10), then and (both are true).
x≡2 (mod 5).

I considered . (set of multiples of a number)