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

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

\(\Rightarrow\) 4(10)+8(1+2+3+...+9)=400.

Rational Numbers | Singapore Mathematics Olympiad, 2013

Try this beautiful problem from Singapore Mathematics Olympiad based on rational numbers.

Problem - Rational Numbers (SMO )

Find the number of positive integer pairs (a,b) satisfying \(a^2 + b^2<2013\) and \(a^{2} b |(b^3 - a^3\)

Key Concepts

Number Theory

Rational Number

Analysis of Numbers

Check the Answer

Answer: 31

Singapore Mathematics Olympiad - 2013 - Senior Section - Problem No. 18

Challenges and Thrills - Pre - College Mathematics

Try with Hints

We can start this sum by rearranging the given values :

Let \( k = \frac { b^3 - a^3 }{a^{2}b} \)

Again we can write it like : \( k = (\frac {b}{a})^2 - \frac {a}{b} \)

Try to use this value and then try to do the rest of the sum.......

From the first hint we can say :

\((\frac {a}{b})^{3} + k (\frac {a}{b})^2 - 1 = 0\)

The only possible positive rational number solution of \(x^3 +kx^2 -1 = 0\) is x = 1 namely a = b . Conversely , if a = b then it is obvious that \(a^2b |(b^3 - a^3\)

Then 2013 > \(a^2 +b^2 = 2a^2 \) implies \(a\leq 31 \). (Answer )

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