Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Find the answer

2. Find all the solutions of each of the following diophantine equations
a) $x^{2}-y^{2}=8$
b) $x^{2}-4 y^{2}=40$
c) $4 x^{2}-9 y^{2}=100$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

2. a) $x= \pm 3, y= \pm 1$
b) no solution
c) $x= \pm 5, y=0 ; x= \pm 13, y= \pm 8$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.