I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$
I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + t(1,-2,0)$ for the homogeneous equation $4x + 2y +5z = 0$ and then I added that to the particular solution to get $(x,y,z) = (50-2s+t,-50-s-2t,2s)$ as the solution. However, Now I need to calculate how many positive solutions. So I have $$50-2s + t > 0$$ $$ -50 -s -2t > 0$$ $$2s > 0$$
I did some algebraic manipulation to get $t > -30$ and $s > 0 $ Now, I still don't understand how to count how many positive solutions are there ?
I went to wolfram alpha and it said $106$ but I have no clue how ? Any Suggestions