![2x+2y+z=20\\ z=\dfrac{20}{2x+2y}\\ z=\dfrac{10}{x+y}](https://tex.z-dn.net/?f=2x%2B2y%2Bz%3D20%5C%5C%0Az%3D%5Cdfrac%7B20%7D%7B2x%2B2y%7D%5C%5C%0Az%3D%5Cdfrac%7B10%7D%7Bx%2By%7D)
Now, for z to be an integer, the sum x+y must be a divisor of 10.
It has to be a positive divisor since z≥0. Also x≠y.
![x+y=1 \\ x=1-y\\ ](https://tex.z-dn.net/?f=x%2By%3D1%20%5C%5C%0Ax%3D1-y%5C%5C%0A)
x≥0 and y≥0 so y can be equal to either 0 or 1. There are 2 solution in this case.
![x+y=2\\ x=2-y\\](https://tex.z-dn.net/?f=x%2By%3D2%5C%5C%0Ax%3D2-y%5C%5C)
In this case, y can be equal to 0,1, or 2, but for y=1 ⇒ x=1, so there are two solutions.
![x+y=5\\ x=5-y](https://tex.z-dn.net/?f=x%2By%3D5%5C%5C%0Ax%3D5-y)
y can be 0,1,2,3,4 or 5 - 6 solutions
![x+y=10\\ x=10-y](https://tex.z-dn.net/?f=x%2By%3D10%5C%5C%0Ax%3D10-y)
y can be 0,1,2,3,4,5,6,7,8,9,10, but for y=5 ⇒ x=5, so 10 solutions.
2+2+6+10=20 solutions in total.