Let d be a positive square free number, h(d) the class number of the field Q(d). In this paper, we prove that: if a,n>2, k>1 are positive integers with 1+4k 2n =da 2 and a<0.9k 25n  or that every prime factor p of n and q of k satisfies (p,q-1)=1. Then h(d)≡0 ( mod n) with the only exception (a,d,k,n)=(5,41,2,4). Meanwhile, we conjecture that the condition (p,q-1)=1 of the above result is not needed.