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

$$\begin{aligned} n\varphi (n)\equiv 2\quad \pmod {\sigma (n)}, \end{aligned}$$

and prove that the only positive integers of the form $2^{\alpha }5^{\beta },\, \alpha , \,\beta \ge 0,$ that satisfy the above congruence are $n=1,\, 2,\, 5,\, 8.$