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Is it possible to build a gapless rectangle with non-integer side lengths using rectangles each with two integer side length and two non-integer side length? The rectangles are not required to be the same size.

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No. What you are asking is not possible. Because if an integer and a non-integer pair up on two opposite sides of the new rectangle, then two of the sides will have non-integer values, but the other two will be forced to have integer values.

Proof:

Given: Rectangle with area of 5N (n is a non-integer number) & Rectangle with area of 4X (x is a non-integer number)

Proven impossible by the following screenshot:

Two long rectangles, sharing a horizontal edge. The top rectangle's top edge is labeled "N" and both side edges are labeled "5". The bottom rectangle's bottom edge is labeled "4" and both side edges are labeled "X". Below the rectangles is "Based on this diagram, N must equal 4, so it would be impossible to make two rectangles with 1 integer and one non-integer value into a rectangle with two sides of non-integer value."

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