Multi-core processors employ shared Last LevelCaches (LLC). This trend will continue in the future withlarge multi-core processors (16 cores and beyond) as well. Atthe same time, the associativity of LLC tends to remain in theorder of sixteen. Consequently, with large multicore processors, the number of cores that share the LLC becomes larger thanthe associativity of the cache itself. LLC management policieshave been extensively studied for small scale multi-cores (4 to8 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 cores. In this study, we introduceAdaptive Discrete and deprioritized Application PrioriTization(ADAPT), an LLC management policy addressing the largemulti-cores where the LLC associativity degree is smaller thanthe number of cores. ADAPT builds on the use of the Footprintnumber 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 cachelines of applications during cache replacement operations. Wefurther find that de-prioritizing certain applications duringcache replacement is beneficial to the overall performance. Weevaluate our proposal on 16, 20 and 24-core multi-programmedworkloads and discuss other aspects in detail.