In MATLAB, the solution is intlinprog. Of course, there may be multiple solutions. intlinprog does not give them, if any could exist. But there is no need for loops either, nor even to go out as far as 100.
Since we know
then we can limit the variables to be no larger than 8.
[x,y,z] = ndgrid(0:8,0:8,0:8);
ind = find(15*x + 16*y + 17*z == 121)
So the only possible solution in positive integers is as found. Fast, efficient, and trivial to write. Sometimes brute force is the easiest thing. Would I have used intlinprog? Of course, as that is the obvious way to solve any problem of this class.
Had the problem been larger, perhaps to find a solution in integers to this?
137*x + 291*y + 313*z + 997*u + 1329*v + 237*w == 1 + 1e15
Now brute force will fail, because the search will push the search space out into numbers on the order of 1e13. And of course, even intlinprog might be at risk, due to the size of the right hand side, compared to the dynamic range of a double. This latter problem can now be solved more easily using number theory. How? Consider this...