how many solutions?

N = 16.

So, 16 positive integer solutions are possible for this equation.

2x + 3y  = 100

3y  =  100 - 2x

y  =  100 - 2x

       _______

             3

We can only have integer solutions for y  whenever  x  = 2 , 5, 8, 11, etc

And the last  x  that  makes y a positive integer  is  when x  = 47

So....the  number  of  integers  =   [ 47 - 2 ] / 3  + 1   =   45 / 3 + 1   =   15  + 1  =  16

Just as Lucky Ducky found   !!!!!

GOOD JOB LUCKYDUCKY  !!!!!