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Unsupervised question answering is an attractive task due to its independence on labeled data. Previous works usually make use of heuristic rules as well as pre-trained models to construct data and train QA models. However, most of these works regard named entity (NE) as the only answer type, which ignores the high diversity of answers in the real world. To tackle this problem, we propose a novel unsupervised method by diversifying answers, named DiverseQA. Specifically, the proposed method is composed of three modules: data construction, data augmentation and denoising filter. Firstly, the data construction module extends the extracted named entity into a longer sentence constituent as the new answer span to construct a QA dataset with diverse answers. Secondly, the data augmentation module adopts an answer-type dependent data augmentation process via adversarial training in the embedding level. Thirdly, the denoising filter module is designed to alleviate the noise in the constructed data. Extensive experiments show that the proposed method outperforms previous unsupervised models on five benchmark datasets, including SQuADv1.1, NewsQA, TriviaQA, BioASQ, and DuoRC. Besides, the proposed method shows strong performance in the few-shot learning setting.