CONO-SUR 2015 Problem 6

Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$ . Two subsets $A$ and $B$ of $S$ are said to be friends if the following conditions are true:

• They do not share any elements.
• They both have the same number of elements.
• The product of all elements from $A$ equals the product of all elements from $B$ .

Prove that there are two subsets of $S$ that are friends such that each one of them contains at least $738$ elements.