Let M ⊂ S n+1 be a complete orientable hypersurface with constant Gauss-Kronecker curvature G. For any v ∈ R n+2 , let us define the following two real functions lv, fv : M → R on M by lv(x) = x, v and fv(x) = ν(x), v with ν : M → S n+1 a Gauss map of M. In this paper, we show that if n = 3, lv = λfv for some nonzero vector v ∈ R 5 and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in the result we just mentioned. We also show that if n = 4, lv = λfv for some nonzero vector v ∈ R 6 and some real number λ and (λ 2 − 1) 2 + (G − 1) 2 = 0, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. Moreover, we will give an example of a complete hypersurface in S 5 with constant Gauss-Kronecker curvature that satisfies the condition lv = λfv for some non zero v, which is neither a totally umbilical hypersurface nor a cartesian product of Euclidean spheres.