Background: When the efficacy of a treatment in a randomized controlled trial is required for multiple primary endpoints, trial design and analysis differ from trial requiring efficacy in only one of the multiple endpoints. Methods: We consider a two-arm clinical trial requiring efficacy analysis for multiple primary endpoints, formulating the appropriate null and alternative hypotheses for the test of treatment efficacy. We study the significance level/statistical power of an intersection-union test (IUT) in this situation. We compare IUT with the intuitive approach (selecting the maximum sample size over those obtained from testing individual primary endpoints one by one) for determination of sample size. Results: The proposed IUT reserves the same Type I error rate as shared by all endpoint-specific tests. The statistical power of the proposed IUT is no more than the minimum from the individual tests. The maximum sample size from multiple end-point-specific tests is often inadequate for the test of treatment efficacy, especially when the standardized effect sizes are similar. Finally, the IUT can be applied to Alzheimer's disease treatment trials in which two primary endpoints are typically used. Conclusions: The IUT is a valid method for use in the design and analysis of clinical trials requiring efficacy at multiple primary endpoints.