+0  
 
0
34
4
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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
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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
avatar+1105 
+9

look at BuilderBoi's answer.

nerdiest  Jul 5, 2022
edited by nerdiest  Jul 8, 2022
 #3
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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
avatar+2275 
+1

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

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