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**Bruno J.** Your notation is terribly confusing. You're saying that $\displaystyle d$ is the order of $\displaystyle a$ mod $\displaystyle r$, and $\displaystyle e$ is the order of $\displaystyle a$ mod $\displaystyle s$, and thus that $\displaystyle lcm(e,d)$ is the order of $\displaystyle a$ mod $\displaystyle rs$?

Hint: use the Chinese remainder theorem. There is an isomorphism $\displaystyle (\mathbb{Z}/rs\mathbb{Z})^* \simeq (\mathbb{Z}/r\mathbb{Z})^* \times (\mathbb{Z}/s\mathbb{Z})^*$.