Multi-core processors employ shared Last Level Caches (LLC). This trend will continue in the future with large multi-core processors (16 cores and beyond) as well. At the same time, the associativity of LLC tends to remain in the order of sixteen. Consequently, with large multicore processors, the number of applications or threads that share the LLC becomes larger than the associativity of the cache itself. LLC management policies have been extensively studied for small scale multi-cores (4 to 8 cores) and associativity degree in the 16 range. However, the impact of LLC management on large multi-cores is essentially unknown, in particular when the associativity degree is smaller than the number of applications. In this study, we introduce Adaptive Discrete and deprioritized Application PrioriTization (ADAPT), an LLC management policy addressing the large multi-cores where the LLC associativity degree is smaller than the number of applications. ADAPT builds on the use of the Footprint-number metric. We propose a monitoring mechanism that dynamically samples cache sets to estimate the Footprint-number of applications and classifies them into discrete (distinct and more than two) priority buckets. The cache replacement policy leverages this classification and assigns priorities to cache lines of applications during cache replacement operations. We further find that de-prioritizing certain applications during cache replacement is beneficial to the overall performance. We evaluate our proposal on 16, 20 and 24-core multi-programmed workloads and discuss other aspects in detail.