+0

# Number Theory

0
78
4

For how many integer values of n between 1 and 1000 inclusive does the decimal representation of n/1375 + n/3 terminate?

Jul 5, 2022

#1
0

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.

Jul 5, 2022
#2
+1155
+9

look at BuilderBoi's answer.

nerdiest  Jul 5, 2022
edited by nerdiest  Jul 8, 2022
#3
0

nerdiest: Did YOU check the answer given as 142 to be correct? Why don't try it and list the 142 numbers.

Guest Jul 5, 2022
#4
+2448
0

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.

Jul 5, 2022