This study explores enhancing grammatical error correction (GEC) through automatic error generation (AEG) using language models (LMs). Specifically, we fine-tune Llama 2 LMs for error generation and find that this approach yields synthetic errors akin to human errors. Next, we train GEC Llama models using these artificial errors and outperform previous state-of-the-art error correction models, with gains ranging between 0.8 and 6 F0.5 points across all tested languages (German, Ukrainian, and Estonian). Moreover, we demonstrate that generating errors by fine-tuning smaller sequence-to-sequence models and prompting large commercial LMs (GPT3.5 and GPT4) also results in synthetic errors beneficially affecting error generation models. We openly release trained models for error generation and correction as well as all the synthesized error datasets for the covered languages.