Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Rational Numbers.
Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.
Integers
Rational Numbers
Euler's Totient Function
But try the problem first...
Answer: is 400.
AIME I, 1992, Question 1
Elementary Number Theory by David Burton
First hint
For Euler's Totient function, there exists 8 numbers that are relatively prime to 30, less than 30.
Second Hint
Here they are in (m,30-m) which in the form of sums of 1
\(\Rightarrow\) sum of smallest eight rational numbers=4
Final Step
there are eight terms between 0 and 1 and there are eight terms between 1 and 2 where these we get as adding 1 to each of first eight terms
\(\Rightarrow\) 4(10)+8(1+2+3+...+9)=400.
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