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Minimum distinct values in a ratio chain

High school · ●●●○○ 50 pts

Let a1,a2,…,a2006a_{1}, a_{2}, \ldots, a_{2006} be positive integers (not necessarily distinct) such that the 2005 ratios a1a2,a2a3,…,a2005a2006\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \ldots, \frac{a_{2005}}{a_{2006}} are all distinct. What is the least possible number of different values among a1,a2,…,a2006a_{1}, a_{2}, \ldots, a_{2006}?

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pigeonhole principlecombinatorial constructionprime factorisation