For how many positive integers n is n/(30n) also a positive integer?

Question

For how many positive integers n is  \frac{n}{30−n}  also a positive integer?

(A) 4 
(B) 5 
(C) 6
(D) 7 
(E) 8

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Archit3110 · Answer

We can plug in values of n to determine ratio is of a +ve integer 
n=29,28,27,25,24,20 15
7 is correct 
Option D

suntprovident · Answer

I think answer would be D as you have not considered n=15

suntprovident · Answer

I solved the question initially by plugging but then realised that there is a simpler way :

Break the numerator to 
1. (30-30+n)/(30-n)
2. Split the fraction : 
(30-n)/(30-n) - 30/(30-n)
3. All nos where 30-n is a factor of 30 and positive will be our solution 
4. N = 15, 20, 24, 25, 27, 28, 29

Alternatively working via brute force and plugging in nos starting from 15 works better

dhart · Answer

Ans should be option D

kevincan · Answer

Also , how about a change in variable: x = 30 - n

n/(30 - n) becomes (30-x)/x = 30/x - 1
 30/x - 1 is a positive integer if and only x is a positive factor of 30 that is less than 30.

30= 2*3*5 has 8 positive factors , 7 of which are less than 30.

For how many positive integers n is n/(30n) also a positive integer?

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GMAT Problem Solving (PS) Questions

For how many positive integers n is n/(30n) also a positive integer?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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