From the given equation in the problem we can deduce that

X=(2Y)/(Y-2)                                            (1)

so we know that

X=[2(Y-2)+4] / (Y-2)=2+4/(Y-2)

Since X must be an integer value 4/(Y-2) must be integer. Therefore Y>2.
Y could  be 6,4 or 3.
Rectangle with  sides 3 and 6 or square with side of 4 satisfy the problem condition (1).  The second condition of the problem excludes the latter option.  So the final answer for the smallest side of the rectangle is 3.

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