AIME 1991 · 第 5 题

Solution

If the fraction is in the form $\frac{a}{b}$, then $a < b$ and $\gcd(a,b) = 1$. There are 8 prime numbers less than 20 ($2, 3, 5, 7, 11, 13, 17, 19$), and each can only be a factor of one of $a$ or $b$. There are $2^8$ ways of selecting some combination of numbers for $a$; however, since $a, only half of them will be between$0 < \frac{a}{b} < 1$. Therefore, the solution is$\frac{2^8}{2} = \boxed{128}$.