Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Problem on Rational Numbers | AIME I, 1992 | Question 1

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


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.

Subscribe to Cheenta at Youtube


Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com