  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 $n=1, 2, 5, 8$.
