<s>[INST] $34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$ There was a number theory question that I have to do for homework.

$34!=295232799cd96041408476186096435ab000000$, find $a$ ,$b$, $c$, and $d$

I know $b=0$ because $10^7\big|34!$ only. But how can I find other variables?
Remark:
No electronic calculation device can be used for calculation. (I need to write the full solution.)
Can someone help me?
 [/INST] Sum of digits is divisible by $9$, which gives
$$a+c+d\equiv5(\mod9).$$
$$2-9+5-2+3-2+7-9+9-c+d-9+6-0+4-1+4-0+8-4+7-6+1-8+6-0+9-6+4-3+5-a\equiv0(\mod11),$$
which gives
$$a+c-d\equiv-1(\mod11).$$
From here we can get $a+c=2$ and $d=3$.
Now, since $a\in\{2,6\}$, we obtain $a=2$, $c=0$ and we are done!
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