For how many integer values of n between 1 and 1000 inclusive does the decimal representation of n/1375 + n/3 terminate?
33 , 66 , 99 , 132 , 165 , 198 , 231 , 264 , 297 , 330 , 363 , 396 , 429 , 462 , 495 , 528 , 561 , 594 , 627 , 660 , 693 , 726 , 759 , 792 , 825 , 858 , 891 , 924 , 957 , 990 , Total = 30 such integers.
+2668 Start with the fraction \({n \over 3}\). This is only terminating when n is a multiple of 3.
Now, look at the \({n \over 1375}\) fraction. This is only terminating when n is a multiple of 11.
So, every \(\text{lcm}(3,11) = 33\) integers, n will terminate.
Thus, there are \(\lfloor {1000 \over 33} \rfloor = \color{brown}\boxed{30}\) integers that work.
