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Let $x$ and $y$ be positive integers such that

\[\frac{1}{x} + \frac{1}{2y} = \frac{1}{6}.\]
What is the smallest possible value of $xy$?

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To find the smallest possible value of $xy$ given the equation $\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}$, we can start by multiplying both sides of the equation by the least common multiple (LCM) of $x$ and $2y$, which is $2xy$. This will help us eliminate the denominators:

$2xy \cdot \left(\frac{1}{x} + \frac{1}{2y}\right) = 2xy \cdot \frac{1}{7}.$

Simplifying, we get:

$2y + x = \frac{2xy}{7}.$

Now, let's rearrange the equation to isolate $x$:

$x = \frac{2xy}{7} - 2y.$

Next, we can factor out $x$ on the right side:

$x = x \cdot \left(\frac{2y}{7}\right) - 2y.$

Dividing both sides of the equation by $x$, we have:

$1 = \frac{2y}{7} - \frac{2y}{x}.$

Simplifying further, we get:

$1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{x}\right).$

Now, let's consider the equation $1 - \frac{2}{x}$. Since $x$ and $y$ are positive integers, the value of $\left(1 - \frac{2}{x}\right)$ must be positive. Therefore, $1 - \frac{2}{x} > 0$, which implies $\frac{2}{x} < 1$.

To minimize the value of $xy$, we want to find the smallest possible positive integer values for $x$ and $y$ that satisfy the given equation. Since $\frac{2}{x} < 1$, the smallest possible value for $x$ is 2.

Substituting $x = 2$ into the equation $1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{x}\right)$, we have:

$1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{2}\right) = \frac{2y}{7} \cdot 0 = 0.$

This implies that $y$ can take any positive integer value. Therefore, the smallest possible value of $xy$ is $x \cdot y = 2 \cdot 1 = 2$.

Hence, the smallest possible value of $xy$ is 2.

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