A variation of a congruence of Subbarao for n = 2 α 5 β

There are many open problems concerning the characterization of the positive integers n fulfilling certain congruences and involving the Euler totient function φ and the sum of positive divisors function σ of the positive integer n. In this work, we deal with the congruence of the form n φ ( n ) ≡ 2 ( mod σ ( n ) ) , and prove that the only positive integers of the form 2 α 5 β , α , β ≥ 0 , that satisfy the above congruence are n = 1 , 2 , 5 , 8 .