Problem:
Find the number of rational numbers , such that when is written as a fraction in lowest terms, the numerator and denominator have a sum of .
Solution:
Let be a rational number in lowest terms with . If , then is also a factor of so is not in lowest terms. On the other hand if and , then , and is in lowest terms. Because , the number of positive integers less than 1000 and relatively prime to is
Half of these numbers (i.e. ) are less than , and these are the possible numerators for a fraction of the desired type.
The problems on this page are the property of the MAA's American Mathematics Competitions