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Unformatted text preview: 1/x + 1/y + 1/z > 1/2 . As x, y, z are primes and 1/5 + 1/7 + 1/11 < 1/2 you conclude that x = 3 . Now yz < 1000/3 so 999 < 2(3y + 3z + yz) < 2(3y + 3z + 1000/3) which gives 55 < y + z . On the other hand 1/6 < 1/y + 1/z , so if y is greater than 5, then z < 42 which makes y + z too small. So y = 5 . The equation for volume gives 15x < 1000 so z < 66 , while the equation for surface area gives 2(15 + 3z + 5z) > 999 so z > 60 . The only option is z = 61 . Brian Laughlin points out there are 29 more solutions if you allow 2 as one of the dimensions. By the way, the number 1 is not considered a prime by (most) mathematicians. this page....
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- Spring '10
- Combinatorics, Prime number, Dilip Chaudhary, Brian Laughlin