> > Is the following conjecture true? > (1): 3^r divides (10^k -1)/9 iff 3^r divides k. > (I verified it to be true for r = 1,2,3 and 4.) About this one, seems to me you can state something more general. Let (m,n,r) be 3 integers > 1 : m==1 (mod n) iff for any r>=0 n^r is the largest power of n dividing (m^(n^r)-1)/(m-1) Benoit Cloitre