Diophantine equation!
Algebra
Level
5
A rectangular prism has dimensions a, b and c and a is not greater than b and b is not greater than c. The dimensions are integers and are greater than 1. How many such prisms are there that satisfy these requirements and have the volume equal to the surface area? Source: AMC 10
10
4
21
12
26
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1 solution
If a is 2 it's not hard to see there isn't a solution. The equation is abc-2ab-2ac-2bc=0 so we see that a=3,4,5,6 and using SFFT on each of those equations we get 5,3,1,1 solutions respectively for a total of 10.