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# Number Theory

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There exist several positive integers x such that 1/(x^2 + 2x) is a terminating decimal. What is the second smallest such integer?

Nov 21, 2021

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We can first factor the denominator as $$\frac{1}{(x)(x+2)}$$. In order for this to be a terminating decimal, both $$x$$ and $$x+2$$ have to only have factors of 2 and/or 5. Testing small values of x, we find that $$x = \{2,8, etc.\}$$. Looking at this, we can see that the second smallest integer value of x is $$\boxed{8}$$

Nov 21, 2021