In this paper, we consider the algorithmic task of content replication and request routing in a distributed caching system consisting of a central server and a large number of caches, each with limited storage and service capabilities. We study a time-slotted system where in each time-slot, a large batch of requests has to be matched to a large number of caches, where each request can be served by any cache which stores the requested content. All requests which cannot be served by the caches are served by fetching the requested content from the central server. The goal is to minimize the transmission rate from the central server. We use a novel mapping between our content replication problem and the Knapsack problem to prove a lower bound on the transmission rate for any content replication policy. Using insights obtained from the mapping, we propose a content replication policy - Knapsack Storage - which achieves this lower bound. While it intuitively makes sense to replicate the more popular contents on a larger number of caches, surprisingly, in certain cases, the Knapsack Storage policy chooses not to replicate the most popular contents on the caches at all.