Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Rational Numbers.
Problem on Rational Numbers – AIME I, 1992
Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.
- is 107
- is 400
- is 840
- cannot be determined from the given information
Check the Answer
But try the problem first…
Answer: is 400.
AIME I, 1992, Question 1
Elementary Number Theory by David Burton
Try with Hints
For Euler’s Totient function, there exists 8 numbers that are relatively prime to 30, less than 30.
Here they are in (m,30-m) which in the form of sums of 1
\(\Rightarrow\) sum of smallest eight rational numbers=4
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
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