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

**Key Concepts**

Integers

Rational Numbers

Euler’s Totient Function

## 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

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.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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