The sum of the reciprocals of two consecutive positive even integers is 5/36. Find the integers.
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Let the two consecutive positive even integers be $x$ and $x+2$. We have $\frac{1}{x}+\frac{1}{x+2}=\frac{5}{36}$. Simplifying the left side, we get $\frac{x+2}{x(x+2)}+\frac{x}{x(x+2)}=\frac{5}{36}$. Combining the fractions on the left side, we have $\frac{x}{x(x+2)}+\frac{2}{x(x+2)}=\frac{5}{36}$. Combining the numerators, we get $\frac{x+2}{x(x+2)}=\frac{5}{36}$. Cross-multiplying, we have $36(x+2)=5x(x+2)$. Expanding both sides, we get $36x+72=5x^2+10x$. Rearranging and combining like terms, we get $5x^2-26x-72=0$. Factoring, we get $(5x+12)(x-6)=0$. Setting each factor equal to 0, we get $5x+12=0$ or $x-6=0$. Solving each equation gives us $x=-\frac{12}{5}$ or $x=6$. Since $x$ must be positive, we have $x=6$. Plugging this back into our original equation, we find that the consecutive positive even integers are $\boxed{6}$ and $\boxed{8}$.
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