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Factorials ending in zero

Source: 2015 AMC 10 B Problem 23

February 26, 2015
factorialmodular arithmeticfloor functionLaTeXAMC

Problem Statement

Let nn be a positive integer greater than 4 such that the decimal representation of n!n! ends in kk zeros and the decimal representation of (2n)!(2n)! ends in 3k3k zeros. Let ss denote the sum of the four least possible values of nn. What is the sum of the digits of ss?
<spanclass=latexbold>(A)</span>7<spanclass=latexbold>(B)</span>8<spanclass=latexbold>(C)</span>9<spanclass=latexbold>(D)</span>10<spanclass=latexbold>(E)</span>11 <span class='latex-bold'>(A) </span>7\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>9\qquad<span class='latex-bold'>(D) </span>10\qquad<span class='latex-bold'>(E) </span>11