DK-GEORGMOHR 2018 Problem 4

A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the $99$ fractions $$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$ ?