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The sum of the reciprocals of two consecutive odd integers is 4/15. Find the integers.
The sum of the reciprocals of two consecutive odd integers is 4/15. Find the integers.
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Let the two consecutive odd integers be $n$ and $n+2$. We set up the equation $\dfrac{1}{n} + \dfrac{1}{n+2} = \dfrac{4}{15}$. To get rid of the fraction, we can multiply both sides by $n(n+2)$, to get $15n + 15(n+2) = 4n(n+2)$. Expanding both sides gives $30n + 30 = 4n^2+ 8n$, so $4n^2 - 22n - 30 = 0$. Next, we can factor out a 2 to get $2(2n^2 - 11n - 15) = 0$, and then factor $(2n-3)(n+5)=(2n-3)(n+5) = 0$. Therefore, $2n-3=0$, so $n=\dfrac{3}{2}$, which isn't an integer; or $n+5=0$, so $n=-5$. Thus, the two integers are $\boxed{-5,-3}$.
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