The Chinese Remainder Theorem · Superchallenge problem

For how many ordered triples of integers $(a, b, c)$ with $0 \le a < 12$, $0 \le b < 18$ and $0 \le c < 20$ does the system $x \equiv a \pmod{12}$, $x \equiv b \pmod{18}$, $x \equiv c \pmod{20}$ have an integer solution $x$?