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This study presents a cross-national quantitative analysis of how university students in the United States and Bangladesh interact with LLMs. Based on an online survey of 318 students, results show LLMs enhance information access, improve writing, and boost academic performance. However, concerns about overreliance, ethical risks, and critical thinking persist. Guided by the AI Literacy Framework, Expectancy-Value Theory, and Biggs' 3P Model, the study finds motivational beliefs and technical competencies shape LLM engagement. Significant correlations emerged between LLM use and perceived literacy benefits (r = .59, p < .001) and optimism (r = .41, p < .001). ANOVA results showed more frequent use among U.S. students (F = 7.92, p = .005) and STEM majors (F = 18.11, p < .001).