If n is a positive interger less than 200 and 14n/60

[condenach]

GMATprep, practice test nº1. Question 11

If n is a positive interger less than 200 and 14n/60 is a positive integer, then n has how many different positive prime factors?

Answers are: 2, 3, 5, 6, 8

The correct one is 3, but I have no idea how to solve this one. Any help?

Thanks

[GMATPaduan]

If n is a positive interger less than 200 and 14n/60 is a positive integer, then n has how many different positive prime factors?

Answers are: 2, 3, 5, 6, 8

I believe the question is asking for the number of distinct positive prime factors

14n/60 can be simplified to 7n/30. If 7n/30 is a positive integer, then 30 must be a factor of n, as it is not a factor of 7.

The possibilities for n (given that n < 200) are 30, 60, 90, 120, 150, 180

If you test these numbers you will quickly see that they all have the same 3 distinct prime factors: 3, 2 and 5.

Hope that helps...

[StaceyKoprince]

Nice explanation, GMATPaduan!

[condenach]

Thanks a lot for such great help

[Aftab]

You don't need to find the prime factors of all the numbers, since they all are multiples of 30. Only finding the prime factors of 30 is good enough.

[AndreaDB]

Hi all.

the simplified denominator is 30 hence n have to be multiple of 30 : this involve that the prime factor are at least 2,3,5 . The next prime could be 7 but 30*7 is 210>200 so the only prime admitted by the question is the triplet 2,3,5 so Three is the right answer.

[JonathanSchneider]

Nicely done.

[ijhsiung]

If I may contribute a similar solution:

Identify out the prime factors of the denominator:
60: 2, 2, 3, 5

Identify the prime factors of the numerator: 2, 7, n

Therefore for the numerator to be divisible, at minimum, it must contain: 2, 3, 5; therefore, 3 different prime positive factors, choice B.

[RonPurewal]

If n is a positive interger less than 200 and 14n/60 is a positive integer, then n has how many different positive prime factors?

Answers are: 2, 3, 5, 6, 8

I believe the question is asking for the number of distinct positive prime factors

14n/60 can be simplified to 7n/30. If 7n/30 is a positive integer, then 30 must be a factor of n, as it is not a factor of 7.

The possibilities for n (given that n < 200) are 30, 60, 90, 120, 150, 180

If you test these numbers you will quickly see that they all have the same 3 distinct prime factors: 3, 2 and 5.

Hope that helps...

i'd like to call extra attention to this particular solution.

it's clear from the answer choices that the actual value of n doesn't matter (since the answers are constants, irrespective of n). therefore, as soon as you find a single value of n that satisfies the hypothesis of the problem, you're done -- just count the prime factors and it's over.

on problems like this, if you don't IMMEDIATELY figure out the theory behind the problem, you should QUICKLY turn to methods like this. it would be a shame to squander several minutes on an unsuccessful attempt at theory, when generating a single value of n is sufficient to solve the problem.