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The sum of the reciprocals of two consecutive positive even integers is 5/36. Find the integers.
The sum of the reciprocals of two consecutive positive even integers is 5/36. Find the integers.
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Koala Tutor
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Let the two consecutive even integers be $n$ and $n+2$. We have the equation
\[\frac{1}{n}+\frac{1}{n+2}=\frac{5}{36}.\]
Multiplying both sides by $36n(n+2)$ gives $36(n+2)+36n=5n(n+2)$. Simplifying and dividing everything by $5$ gives $7n+9=n^2+2n$. Bringing everything to one side gives $n^2-3n-9=0$. We can either factor this as $(n+1)(n-9)=0$ or use the quadratic formula to find $n=9$, because $n$ is clearly positive. Then, the two consecutive positive even integers are $\boxed{8,10}$.
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