LSTMs trained on next-word prediction can accurately perform linguistic tasks that require tracking long-distance syntactic dependencies. Notably, model accuracy approaches human performance on number agreement tasks (Gulordava et al., 2018). However, we do not have a mechanistic understanding of how LSTMs perform such linguistic tasks. Do LSTMs learn abstract grammatical rules, or do they rely on simple heuristics? Here, we test gender agreement in French which requires tracking both hierarchical syntactic structures and the inherent gender of lexical units. Our model is able to reliably predict long-distance gender agreement in two subject-predicate contexts: noun-adjective and noun-passive-verb agreement. The model showed more inaccuracies on plural noun phrases with gender attractors compared to singular cases, suggesting a reliance on clues from gendered articles for agreement. Overall, our study highlights key ways in which LSTMs deviate from human behaviour and questions whether LSTMs genuinely learn abstract syntactic rules and categories. We propose using gender agreement as a useful probe to investigate the underlying mechanisms, internal representations, and linguistic capabilities of LSTM language models.