A variation of a congruence of Subbarao for \(n=2^\alpha 5^\beta\) (Q682143)

Connecting to several related results in the literature, the author proves that all solutions of the congruence \(n\varphi(n)\equiv 2\pmod{\sigma(n)}\) of the form \(n=2^\alpha 5^\beta\) \((\alpha\geq 0,\beta\geq 0)\) are given by \(n=1,2,5,8\). In the proofs, besides elementary congruences and the properties of the functions involved, the theory of Pellian equations is also applied.