Cultural competence, defined as the ability to understand and adapt to multicultural contexts, is increasingly vital for large language models (LLMs) in global environments. While several cultural benchmarks exist to assess LLMs' cultural competence, current evaluations suffer from fragmented taxonomies, domain specificity, and heavy reliance on manual data annotation. To address these limitations, we introduce CultureSynth, a novel framework comprising (1) a comprehensive hierarchical multilingual cultural taxonomy covering 12 primary and 130 secondary topics, and (2) a Retrieval-Augmented Generation (RAG)-based methodology leveraging factual knowledge to synthesize culturally relevant question-answer pairs. The CultureSynth-7 synthetic benchmark contains 19,360 entries and 4,149 manually verified entries across 7 languages. Evaluation of 14 prevalent LLMs of different sizes reveals clear performance stratification led by ChatGPT-4o-Latest and Qwen2.5-72B-Instruct. The results demonstrate that a 3B-parameter threshold is necessary for achieving basic cultural competence, models display varying architectural biases in knowledge processing, and significant geographic disparities exist across models. We believe that CultureSynth offers a scalable framework for developing culturally aware AI systems while reducing reliance on manual annotation\footnote{Benchmark is available at https://github.com/Eyr3/CultureSynth.}.