A variation of a congruence of Subbarao for $$n=2^{\alpha }5^{\beta }$$

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 \(n=1,\, 2,\, 5,\, 8.\)