Given that

34! = 295 232 799 cd9 604 140 847 618 609 643 5ab 000 000,

determine the digits a, b, c, d.

so far i know that b = 0 as 34! has a factor of $\displaystyle 5^7$

next what i thought of doing was taking the digital root (excluding c d and a) giving me 139 or 4mod9 or 1mod3

a+c+d must equal 5mod9 and 2mod3 (which sadly are the exact same thing)

because the number also has a facotor of 11 so taking the alt digital root gives me 21 +d -a -c given d - a -c is 1mod11.

Okay I'm not totally sure about what to do from here. some help please.

p.s. if possible could someone reconmend some good reading on the topic of modular arithmetic. thanks