Problem

Source: Czech-Polish-Slovak Junior Match 2014, Team p1 CPSJ

Tags: number theory, greatest common divisor, Subsets, Product



The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?