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Re: HOW MANY different positive prime factors? [#permalink]

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29 Jul 2009, 10:56

14n/60 = 7n/30 since this is an integer n has to be a multiple of 30. now, the prime factors of 30 = 2, 3, 5 hence n must have atleast these 3 prime factors.

I like to put the numbers in prime factors so it is easier and faster to visualize.

14*n/60

If we write the factors of 14 --> 2, 7, and the factors of 60 --> 2, 2, 3, 5, we have

(2*7*n)/(2^2*3*5) Simplifying 7*n/(2*3*5)

The only way the equation above has an integer value is if n has AT LEAST the factors 2, 3 and 5, so we can simplify again and we have the number 7.

The number could be 2*3*5, or 2*3*5*2, or 2*3*5*.....

However to be less than 200 we can not add any prime number. 2*3*5 = 120 If we added the next prime factor 7, we would have 2*3*5*7 = 840

Thus, answer B

Again, The explanation might looks long. But try to do this question following this procedure and you will see it is faster.

Good studies PS.: If you liked the explanation, consider a kudo! I am almost accessing the GMATClub tests!!!

2*3*5=30 & its not 120.... or I have missed something here.. Kindly explain

Clearly it's a typo.

In N is a positive integer less than 200, and 14N/60 is an integer, then N has how many different positive prime factors?

A. 2 B. 3 C. 5 D. 6 E. 8

Given: \(0<n=integer<200\) and \(\frac{14n}{60}=integer\).

\(\frac{14n}{60}=\frac{7n}{30}=integer\) --> \(\frac{7n}{30}\) to be an integer \(n\) must be a multiple of \(30=2*3*5\), so \(n\) definitely has these three different positive prime factors. Also, \(n\) can not have more than 3 as if it has for example 4 different prime factors then least value of \(n\) would be \(2*3*5*7=210>200\).

So \(n\) has exactly 3 different positive prime factors: 2, 3, and 5.

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