Denote by Z(p) (resp.(Z)p) the p localization (resp.p completion) of Z.Then we have the canonical inclusion Z(p) (→) (Z)p.Let S2n-1(p) be the p-local (2n-1)-sphere and let B2n(p) be a connected p-local space satisfying S2n-1(p) (≌) ΩB2n(p); then H*(B2n(p),Z(p)) =Z(p)[u] with |u| =2n.Define the degree of a self-map f of B2n(p) to be k ∈ Z(p) such that f*(u) =ku.Using the theory of integer-valued polynomials we show that there exists a self-map of B2n(p) of degree k if rand only if k is an n-th power in (Z)p.