1991 AIME Problems/Problem 5
Problem
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be the resulting product?
Solution
If the fraction is in the form , then
and
. There are 8 prime numbers less than 20 (
), and each can only be a factor of one of
or
. There are
ways of selecting some combination of numbers for
; however, since
, only half of them will be between
. Therefore, the solution is
.
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |