I solved Problem 828 which is a much harder version of this problem, and then I remembered that I haven't solved this one yet, and it was my last problem out of the first 100!
I just copied my recursive code (See my explanation of problem 828), made a one line difference to include fractions, then I just find all numbers I can make, keep only the integers and test for a streak. Code runs in 7ms
Input 4 integers separated by a space (a, b, c, d)
Code will output the the longest streak of numbers and all the constructable numbers using (a, b, c, d)