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Kinda Partitiony NT

Source: AMC 12 2006B, Problem 22

February 17, 2006
floor functionfunctioninequalitiesalgebraAMC

Problem Statement

Suppose a,b, a, b, and c c are positive integers with a \plus{} b \plus{} c \equal{} 2006, and a!b!c! \equal{} m\cdot10^n, where m m and n n are integers and m m is not divisible by 10. What is the smallest possible value of n n? <spanclass=latexbold>(A)</span>489<spanclass=latexbold>(B)</span>492<spanclass=latexbold>(C)</span>495<spanclass=latexbold>(D)</span>498<spanclass=latexbold>(E)</span>501 <span class='latex-bold'>(A) </span> 489 \qquad <span class='latex-bold'>(B) </span> 492 \qquad <span class='latex-bold'>(C) </span> 495 \qquad <span class='latex-bold'>(D) </span> 498 \qquad <span class='latex-bold'>(E) </span> 501