Proof Writing & Problem Solving · Hard problem

A proof about an integer $n$ splits into cases according to the three remainders of $n$ on division by $4$, by $6$ and by $9$. The author lists one case for each triple of remainders, but only triples that some integer $n$ actually has need a case. Of the $4 \cdot 6 \cdot 9 = 216$ triples, how many actually occur?