2005 iTest Problems/Problem 25

Problem

Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$ . How many elements of this set are divisible by $2005$ ?

Solution 1

If for any $N$ , $N!$ is divisible by 2005, any integers greater than $N$ also share that property. So we should start by finding the smallest factorial that is divisible by 2005. Since $2005=5\cdot401$ , and 401 is prime, the smallest such $N$ is 401, as no smaller factorials are divisible by 401. Thus there are $401-1=400$ factorials that are not divisible by 2005, and our answer is $2005-400=\boxed{1605}$ such elements.

2005 iTest (Problems, Answer Key)
Preceded by: Problem 24	Followed by: Problem 26
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2005 iTest Problems/Problem 25

Problem

Solution 1

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