2x + 3y + 12z = 180

2x = 180 - 3y - 12z

2x = 3(60 - y - 12z)

Let positive integer 60-y-12z = A then

So 2x = 3A
   2x = 2A + A
    x = A + A/2
x - A = A/2

Since x - A is a positive integer, so is A/2, say B, and A = 2B

x - A = A/2
x - 2B = 2B/2
x - 2B = B
     x = 3B

---------------

2x + 3y + 12z = 180

3y = 180 - 2x - 12z

3y = 2(90 - x - 6z)

Let positive integer 90-x-6z = C then

So 3y = 2C
    y = 2C/3
    
Since y is a positive integer C must be divisible by 3, 
so C = 3D, so

   y = 2(3D)/3
   y = 2D 

--------------------------------
So we substitute x = 3B and y = 2D into the original eqution:

2x + 3y + 12z = 180

2(3B) + 3(2D) + 12z = 180
6B + 6D + 12z = 180
 B + D + 2z = 30
         2z = 30 - (B + D)

The largest value 2z can take on is when B and D are both as small
as can be, which is 1 each, so

         2z <= 30 - (1 + 1)
         2z <= 28
          z <= 14

So we know that z can take on any integer value from 1 through 14

 B + D = 30 - 2z
     B = 30 - 2z - D


For any of those 14 values of z, B can be chosen anywhere from 1
through when D is the smallest value 1, or (30 - 2z - 1) or (29 - 2z) 
 
Therefore the number of solutions is

, which equals

, which equals



The sum of the first n integers is  so
the sum of the first 14 integers is ,
so we can replace the summation by 105



So the number of positive integer solutions is 196.

Edwin