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)