A positive integer is twice another. The sum of the reciprocals of the two positive integers is 3/14. Find the two integers.
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Let's call the smaller of the two integers $x$, and the other integer $2x$ (since it is twice the value of the smaller integer).
The first equation we can write is $2x = x + x$, which simply expresses that the larger integer is twice the value of the smaller integer.
The second equation can be written as $\frac{1}{x} + \frac{1}{2x} = \frac{3}{14}$.
To solve this equation, we can get rid of the fractions by multiplying both sides of the equation by $14x$:
\[14x \left(\frac{1}{x} + \frac{1}{2x}\right) = 14x\left(\frac{3}{14}\right).\] Simplifying the left side gives:
\[14x \cdot \frac{1}{x} + 14x \cdot \frac{1}{2x} = 3.\] This simplifies to:
\[14 + 7 = 3,\] which is not true.
Therefore, there are no positive integers that satisfy both equations.
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