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How do neural language models keep track of number agreement between subject and verb? We show that `diagnostic classifiers', trained to predict number from the internal states of a language model, provide a detailed understanding of how, when, and where this information is represented. Moreover, they give us insight into when and where number information is corrupted in cases where the language model ends up making agreement errors. To demonstrate the causal role played by the representations we find, we then use agreement information to influence the course of the LSTM during the processing of difficult sentences. Results from such an intervention reveal a large increase in the language model's accuracy. Together, these results show that diagnostic classifiers give us an unrivalled detailed look into the representation of linguistic information in neural models, and demonstrate that this knowledge can be used to improve their performance.