The dimensions of the rectangular prism are x, x, and y.
The volume is
The surface area is
We need to find the number of solutions in positive integers of
Divide out the common factor x. This is valid as long as x is not 0; and x=0 is not possible in this problem.
Solve the equation for one variable in terms of the other.
In that final form, 2 and y are integers, so has to be an integer.
The problem only asks for the number of solutions; the answer is the number of positive integral factors of 8, which is 4 (1, 2, 4,and 8).
The problem is more interesting if we continue to find the four solutions and verify that they are solutions.
Make a table showing the solutions and verifying that they are solutions, using the four positive integral factors of 8:
x-4 x 8/(x-4) y x^2y 2x^2+4xy
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1 5 8 10 25*10 = 250 2(25)+4(5)(10) = 50+200 = 250
2 6 4 6 36*6 = 216 2(36)+4(6)(6) = 72+144 = 216
4 8 2 4 64*4 = 256 2(64)+4(8)(4) = 128+128 = 256
8 12 1 3 144*3 = 432 2(144)+4(12)(3) = 288+144 = 432