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34!=295232799cd96041408476186096435ab000000 Find $a,b,c,d$ (all single digits).

BMO 2002

Number Theory
Easy
Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

Get prepared to find the residue of 34! modulo various divisors! The substitution $x=\frac{34!}{10^7}=295232799cd9604140809643ab$ should help simplify the calculation.
Note that $x$ is divisible by $2^7$. Find all possible residues of a number modulo $10^7$ given that the number is divisible by $2^7$. That’ll help you prove that $a=5, b=2$.
Note that $x$ is divisible by 9. This gives that $c+d$ is either 3 or 12.
As $x$ is also divisible by 11, we must have $c-d=-3$ or 8. As $2c$ has to be an even integer less than or equal to 18, we must have $c+d=3, c-d=-3$. This gives $c=0, d=3$.

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