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Only 1 rectangle can be built with a Prime number of square tiles.
If you can only make one rectangle, your number's prime. If you can make more than one, it's composite.
How long is the rectangle? If the rectangle is only one tile long, then it will take 20 tiles.
Two tiles can only make one rectangle.
No single integer satisfies both requests. 80 is the only integer with exactly 5 factor pairs in that range, but it has two prime factors. 64 and 81 have one prime factor but 4 and 3 factor pairs respectively.