1992 AIME Problems/Problem 1
Problem
Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.
Solution
Solution 1
There are 8 fractions which fit the conditions between 0 and 1:
Their sum is 4. Note that there are also 8 terms between 1 and 2 which we can obtain by adding 1 to each of our first 8 terms. For example, Following this pattern, our answer is
Solution 2
By Euler's Totient Function, there are numbers that are relatively prime to , less than . Note that they come in pairs which result in sums of ; thus the sum of the smallest rational numbers satisfying this is . Now refer to solution 1.
Solution 3
Note that if is a solution, then is a solution. We know that Therefore the answer is
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