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Infinitely many.

Select any number L, greater than sqrt(24) units an let B = 24/L.

then the rectangle with sides measuring L and B will have an area of L*B = L*24/L = 24 square units.

Also, B ≤ sqrt(24) ≤ L so that value of L gives a different rectangle. And, since there are infinitely many possible values for L, there are infinitely many possible rectangles.

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Q: How many different rectangles can you make with a area of 24?
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