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

Abstract

There are many open problems concerning the characterization of the positive integers $n$ fulfilling certain congruences and involving the Euler totient function $\varphi$ and the sum of positive divisors function $\sigma$ of the positive integer $n$. In this work, we deal with the congruence of the form $$ n\varphi(n)\equiv2\pmod{\sigma(n)} $$ and we prove that the only positive integers of the form $2^{\alpha}5^{\beta}, \enspace \alpha, \beta\geq0,$ that satisfy the above congruence are $n=1, 2, 5, 8$.