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# Understand the problem

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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#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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