In N is a positive integer less than 200, and 14N/60 is an : GMAT Problem Solving (PS)
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# In N is a positive integer less than 200, and 14N/60 is an

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In N is a positive integer less than 200, and 14N/60 is an [#permalink]

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27 Jul 2009, 00:15
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-n-is-a-positive-integer-less-than-200-and-14n-60-is-an-100763.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 03 Nov 2012, 00:24, edited 2 times in total.
Renamed the topic and edited the question.
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Re: HOW MANY different positive prime factors? [#permalink]

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27 Jul 2009, 00:32
X = 14n/60 is an integer

X= 14n/60 = 7n/30

n has to be a multiple of 30 in this case for X to be integer.

possible values for n < 200
= 30*(1 to 6)

==> possible prime factors - 2,3 and 5
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Re: HOW MANY different positive prime factors? [#permalink]

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27 Jul 2009, 16:23
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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

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

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29 Jul 2009, 09:56
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.

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

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03 Nov 2012, 00:17
coelholds wrote:
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

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

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03 Nov 2012, 00:25
Expert's post
1
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BOOKMARKED
breakit wrote:
coelholds wrote:
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

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.

Hope it helps.

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-n-is-a-positive-integer-less-than-200-and-14n-60-is-an-100763.html
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Re: HOW MANY different positive prime factors?   [#permalink] 03 Nov 2012, 00:25
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