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 and will be the resulting product?
Solution:
For a fraction to be in lowest terms, its numerator and denominator must be relatively prime. Thus any prime factor that occurs in the numerator cannot occur in the denominator, and vice-versa. There are eight prime factors of , namely , and . For each of these prime factors, one must decide only whether it occurs in the numerator or in the denominator. These eight decisions can be made in a total of ways. However, not all of the resulting fractions will be less than . Indeed, they can be grouped into pairs of reciprocals, each of which will have exactly one fraction less than . Thus the number of rational numbers with the desired property is .
The problems on this page are the property of the MAA's American Mathematics Competitions