State True or False. The number of elements in the set $$\{x \mid x(x + 8)(x + 58) = 2013, x \text{ is an integer}\}$$ is 1.

Correct Answer: True

Solution

$$x = 3$$ works since $$3 \times 11 \times 61 = 2013$$. The equation is $$x^3 + 66x^2 + 464x - 2013 = 0$$, and dividing by $$x - 3$$ leaves $$x^2 + 69x + 671 = 0$$, whose discriminant $$4761 - 2684 = 2077$$ is not a perfect square. So $$x = 3$$ is the only integer solution and the statement is True.

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