The power of factorials

Computer Science Level 4

Determine the sum of all non-negative integers n n such that n ! = m × 1 0 n , 1 ≤ m < 10 n! = m \times 10^n, 1\leq m < 10 .


The answer is 51.

Miles Koumouris by Miles Koumouris , 20, Australia

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1 solution

Miles Koumouris
Jan 1, 2017

Note that

0 ! = 1 × 1 0 0 25 ! ≈ 1.5511 × 1 0 25 26 ! ≈ 4.0329 × 1 0 26 , \begin{aligned} 0! &= 1\times 10^0\\ 25!&\approx 1.5511\times 10^{25}\\ 26!&\approx 4.0329\times 10^{26},\\ \end{aligned} and since 27 > 10 27>10 , the answer must be 0 + 25 + 26 = 51 0+25+26=\boxed{51} .

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