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: $\frac{1}{30},\frac{7}{30},\frac{11}{30},\frac{13}{30},\frac{17}{30},\frac{19}{30},\frac{23}{30},\frac{29}{30}$

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, $1+\frac{19}{30}=\frac{49}{30}.$ Following this pattern, our answer is $4(10)+8(1+2+3+\cdots+9)=\boxed{400}.$

Solution 2

By Euler's Totient Function, there are $8$ numbers that are relatively prime to $30$ , less than $30$ . Note that they come in pairs $(m,30-m)$ which result in sums of $1$ ; thus the sum of the smallest $8$ rational numbers satisfying this is $\frac12\cdot8\cdot1=4$ . Now refer to solution 1.

Solution 3

Note that if $x$ is a solution, then $(300-x)$ is a solution. We know that $\phi(300) = 80.$ Therefore the answer is $\displaystyle\frac{80}{2} \cdot \displaystyle\frac{300}{30} = \boxed{400}.$

1992 AIME (Problems • Answer Key • Resources)
Preceded by First question	Followed by Problem 2
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All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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