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Estonia algebra

Problem

Teacher tells Jüri two nonzero integers and such that is divisible by . Jüri has to find a nonzero integer such that is divisible by and all solutions of the quadratic equation are integers. Can Jüri always solve the problem?
Solution
By the conditions of the problem there is an integer such that . Let ; then and is divisible by . The quadratic equation or has solutions and .
Final answer
Yes; for b = a q, choose c = -2 a q^2. Then the roots are q and -2 q.

Techniques

Vieta's formulasIntegers