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There are 34 rectangular integers less than 100 which are the product of two primes. Can a room be covered precisely (no overlaps, etc.) with the 34 rectangles that have as areas these products?

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No.

The total area of the rectangles is $1707= 3 \cdot 569$. That rectangle is too narrow for the carpets of width $5$ or $7$

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  • $\begingroup$ And if we exclude the four square rectangles and thus are left with total area 1620? $\endgroup$ Nov 1, 2022 at 18:59
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    $\begingroup$ That is much harder. You need one dimension larger than $47$, so if it works I would bet on $60 \times 27$ $\endgroup$ Nov 1, 2022 at 19:20
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Here's a tiling for the suggested alternative, using the 30 rectangles with distinct prime sides and area less than 100, to tile a 60x27 rectangle. Just brute force but I helped a bit by ordering the rectangles 'longest first'.enter image description here

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