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Here's an answer that would have been a lot easier to write if MathJax had been available:

• Is there any more optimal way to solve this idempotent equation?

  So if n can be factored as p₁^e₁ . p₂^e₂ . ... then each p_i^e_i must divide either a or a − 1 (because each p_i can divide into at most one of a and a − 1).