Problem on Rational Numbers | AIME I, 1992

Content

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.

Integers

Rational Numbers

Euler’s Totient Function

But try the problem first…

Answer: is 400.

Source

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.