The sets \(A = \{z : z^{18} = 1\}\) and \(B = \{w : w^{48} = 1\}\) are both sets of complex roots of unity. The set \(C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}\) is also a set of complex roots of unity. How many distinct elements are in \(C^{}_{}\)?

(第八届AIME1990 第10题)