## Understand the problem

34!=295232799cd96041408476186096435ab000000 Find (all single digits).

##### Source of the problem

BMO 2002

##### Topic

Number Theory

##### Difficulty Level

Easy

##### Suggested Book

An Excursion in Mathematics

## Start with hints

Do you really need a hint? Try it first!

Get prepared to find the residue of 34! modulo various divisors! The substitution should help simplify the calculation.

Note that is divisible by . Find all possible residues of a number modulo given that the number is divisible by . That’ll help you prove that .

Note that is divisible by 9. This gives that is either 3 or 12.

As is also divisible by 11, we must have or 8. As has to be an even integer less than or equal to 18, we must have . This gives .

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