The reason that the alt digital root method works for factors of 11 is that , so that , and so on. There is a similar (slightly more elaborate) technique for multiples of 7 or 13, based on the fact that 1001=7×11×13. This means that , so that and . The same thing is true with 13 in place of 7.
To apply this technique to 34!, carry out the alt digital root procedure just as in the case of 11, but using groups of three digits rather than just single digits. This tells you that 295232+799cd9+604140+847618+609643+5a0 is a multiple of 7 (and also of 13). That gives you two more equations for a, c and d.