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Number Theory Level 3

Find the sum of all positive integers n n such that n ! n! ends with exactly
1000 1000 zeroes.

20035 16445 No such number exists 16418 None of the above

Priyanshu Mishra by Priyanshu Mishra , 20, India

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1 solution

Priyanshu Mishra
Nov 8, 2015

We can use legendre's formula i.e.,

e p ( n ) = n − S p ( n ) n − 1 \large\ { e }_{ p }(n) = \frac { n - S_{ p }(n) }{ n - 1 }

Which implies that n ≥ 4000 n \ge 4000 . Jumping 5 times and introspecting we get,

4005 ≤ n ≤ 4009 4005 \le n \le 4009 giving the sum to be 20035 \boxed{20035} .

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