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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 $a^{2} b |(b^3 - a^3$

• 30
• 32
• 31
• 29

Key Concepts

Number Theory

Rational Number

Analysis of Numbers

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 )