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

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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

Originally posted by DSGB on 09 Sep 2010, 16:39.
Last edited by bb on 17 Sep 2018, 23:26, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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jjewkes wrote:
I am not sure how to solve this one:

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.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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"n" has to be multiple of 60 if 14n/60 has to be an integer. The multiples of 60 < 200 are 60,120,180.

The prime factors of 60 are 2,3,5 . It would be the same with 120 and 180.
The number of prime factors are 3. Hence B.

Pls correct me if my explanation is wrong.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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dharani1234 wrote:
"n" has to be multiple of 60 if 14n/60 has to be an integer. The multiples of 60 < 200 are 60,120,180.

The prime factors of 60 are 2,3,5 . It would be the same with 120 and 180.
The number of prime factors are 3. Hence B.

Pls correct me if my explanation is wrong.

For 14n/60 to be an integer, n only needs to be a multiple of 30.
But the same logic holds even in that case.
As a multiple of 30, it already has 2,3,5 as factors
For 7 to be a factor (next smallest prime), n would need to be 210 which is not possible
Hence the answer is 3.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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shrouded1 wrote:
dharani1234 wrote:
"n" has to be multiple of 60 if 14n/60 has to be an integer. The multiples of 60 < 200 are 60,120,180.

The prime factors of 60 are 2,3,5 . It would be the same with 120 and 180.
The number of prime factors are 3. Hence B.

Pls correct me if my explanation is wrong.

For 14n/60 to be an integer, n only needs to be a multiple of 30.
But the same logic holds even in that case.
As a multiple of 30, it already has 2,3,5 as factors
For 7 to be a factor (next smallest prime), n would need to be 210 which is not possible
Hence the answer is 3.

Yeah I think my answer would have been wrong if the question was n<220 ... in which case I would have considered 30 Magoosh GMAT Instructor Joined: 28 Nov 2011
Posts: 300
Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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5
The lowest prime number, 2, is also the only even prime number. 2, 3, 5, 7, 11…

Every number is made up of at least one prime factor, except for the number 1.

Every number that is not a prime can be broken into prime factors.

Of course this is very basic, but it is important to keep in mind when factoring larger numbers. For instance if we take the number 60 we can see that it is comprised of the factors 4 x 3 x 5. Notice that 3 and 5 are both primes, however 4 is not. A number that is not a prime can always be broken down into more than one prime number, whether those number or numbers are distinct. Therefore 4 can be broken down to 2 and 2.

When taking apart larger numbers sometimes a factoring tree can be helpful. (The U.S. emphasizes this skill and thus it comes naturally for those schooled in the U.S. ). With a large number sometimes the easiest way to approach it is by dividing by 2 if it is even, and if odd, knowing the divisibility rules for 3, 5, etc.

Let’s take a random number, say 136. We can start dividing by 2 as follows: 136/2 = 68, 68/2 = 34, 34/2 = 17. Now we have the prime factors. Three ‘2s’ and a ’17.’ Sometimes a question, such as the question in the thread, will ask for distinct or different primes. In the case of 136, the distinct primes will be 2 and 17.

This of course is really high-level and unless you are at the 200-300 GMAT level you would never such a question. Nonetheless, these fundamentals apply even to difficult prime factorization problems.

So back to the question at hand:

14n/60, can be reduced to 7n/30. Because 7n/30 has to be an integer, n has to be a multiple of 30. The prime factors of 30 are 2, 3, and 5.

The next important part to the question is “different positive factors.” So if we multiply the prime factors 2, 3, 5 times 30, we do not change the number of different prime factors. But as soon as we multiply n times the next highest prime factor, 7, we go over 200: n = 30x7 = 210. Therefore n contains only three prime factors: 2, 3, and 5.

Hope that helped!
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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7n/30 tells us that n is a factor of 30, which has 2, 3, and 5 as its prime factors. Each of them is distinct. Moreover, all multiples of 30 less than 200 can be derived by multiplying these prime factors alone. Thus, number of different prime factors is 3.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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60 can be factorized into 2,2,3 and 5.
14 already has one 2 so n must have 2,3 and 5 to yield an integer and which are 3 distinct prime numbers hence B.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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14n/60, lets factorize this as 2x7xn / 2x2x3x5 -> 7xn / 2x3x5. Therefore, n has 2, 3, 5 for this to be an integer. That is n needs to be a factor of 30.

3 prime factors, B.
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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Impenetrable wrote:
Hi guys,
I never really had to deal with prime factors (they don't really matter in the German school system) and that's why I often struggle with questions.

In n is a positive integer smaller than 200 and (14n)/60 is an integer, then n has how many different positive prime factors?

2
3
5
6
8

Can you please explain your answer and maybe give me some tipps for those kind of questions?

Thanks!

Here first we further break down the no 14n/60 to 7n/30

now 30 = 2*3*5

so to get an integer the numerator must be a multiple of 2*3*5
so the smallest number which satisfies this condition is 30
thus we have 3 prime numbers here for n i.e 2,3,5

so ans is B
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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Impenetrable wrote:
Hi guys,
I never really had to deal with prime factors (they don't really matter in the German school system) and that's why I often struggle with questions.

In n is a positive integer smaller than 200 and (14n)/60 is an integer, then n has how many different positive prime factors?

2
3
5
6
8

Can you please explain your answer and maybe give me some tipps for those kind of questions?

Thanks!

Some time back, I had written a couple of posts on prime factors discussing their usage on GMAT. You might find them useful.

http://www.veritasprep.com/blog/2010/12 ... ly-number/
http://www.veritasprep.com/blog/2010/12 ... t-squares/
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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I multiplied 60 x 14 and did prime factorisation, counted the prime #´s from 200 below and was 3, 5 and 7 = 3 prime #´s. Can it be solved that way or I was lucky?
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GRE 1: Q169 V154 Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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Here 14n/60 is an integer
Hence n=30p for some p
n<200=>n can be 30,60,90,120,150,180
All have three prime factors
Smash that B
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Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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Impenetrable wrote:
In n is a positive integer smaller than 200 and (14n)/60 is an integer, then n has how many different positive prime factors?

2
3
5
6
8

$$\frac{14n}{60}$$ = $$\frac{2*7*n}{2^2*3*5}$$ = $$\frac{7*n}{2*3*5}$$

Since the numerator and denominator has nothing in common so, n must be a multiple of 2*3*5 for the numerator to be divisible by 60

Thus the number of Prime factors of n will be 3

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Originally posted by Abhishek009 on 21 Oct 2016, 11:21.
Last edited by Abhishek009 on 22 Oct 2016, 02:15, edited 1 time in total.
TYPO edited : a multiple of
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GRE 1: Q169 V154 Re: If N is a positive integer less than 200, and 14N/60 is an  [#permalink]

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Abhishek009 wrote:
Impenetrable wrote:
In n is a positive integer smaller than 200 and (14n)/60 is an integer, then n has how many different positive prime factors?

2
3
5
6
8

$$\frac{14n}{60}$$ = $$\frac{2*7*n}{2^2*3*5}$$ = $$\frac{7*n}{2*3*5}$$

Since the numerator and denominator has nothing in common so, n must be 2*3*5 for the numerator to be divisible by 60
Thus the number of Prime factors of n will be 3

=> Hey..! Looks like there is an error here Abhishek009.You shouldn't write that n must be 2*3*5. Instead N can be 2*3*5 or 2*3*5*2 or 2*3*5*3 or 2*3*5*4 or 2*3*5*5 . In all the cases N will have 3 prime factors.
The smallest value for n to have more than 3 prime factors will be 210 which is not allowed in the bound specified for n

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

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stonecold wrote:
Abhishek009 wrote:
Impenetrable wrote:
In n is a positive integer smaller than 200 and (14n)/60 is an integer, then n has how many different positive prime factors?

2
3
5
6
8

$$\frac{14n}{60}$$ = $$\frac{2*7*n}{2^2*3*5}$$ = $$\frac{7*n}{2*3*5}$$

Since the numerator and denominator has nothing in common so, n must be a multiple of 2*3*5 for the numerator to be divisible by 60
Thus the number of Prime factors of n will be 3

=> Hey..! Looks like there is an error here Abhishek009.You shouldn't write that n must be 2*3*5. Instead N can be 2*3*5 or 2*3*5*2 or 2*3*5*3 or 2*3*5*4 or 2*3*5*5 . In all the cases N will have 3 prime factors.
The smallest value for n to have more than 3 prime factors will be 210 which is not allowed in the bound specified for n

It was a TYPO in the highlighted part, else it will change our answer as you have correctly pointed out... _________________
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Abhishek....

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

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DSGB wrote:
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

We are given that N is a positive integer less than 200, and 14N/60 is an integer, and we need to determine the number of different positive prime factors of N. Let’s begin by simplifying 14N/60.

14N/60 = 7N/30

In order for 7N/30 to be an integer, N must be divisible by 30. In other words, N must be a multiple of 30. The multiples of 30 less than 200 are: 30, 60, 90, 120, 150 and 180. First let’s investigate 30, the smallest positive number that is a multiple of 30.

Since 30 = 2 x 3 x 5, N has 3 different positive prime factors.

However, even if we break 60, 90, 120, 150, or 180 into prime factors, we will see that each of those numbers has 3 different prime factors (2, 3, and 5).

Thus, we can conclude that N has 3 different positive prime factors

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

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DSGB wrote:
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

Let's choose a nice value of N that satisfies the given information.

GIVEN: 14N/60 is an integer
Prime factorize the numerator and denominator to get: (7)(2)(N)/(2)(2)(3)(5) is an integer
Simplify: (7)(N)/(2)(3)(5) is an integer
Notice that, when N = 30 (aka the product of 2, 3, and 5), then (7)(N)/(2)(3)(5) = (7)(2)(3)(5)/(2)(3)(5) = 7, which IS an integer
So, N = 30 satisfies the given information.

N has how many different positive prime factors?
30 = (2)(3)(5)
So, N has 3 different positive prime factors (2, 3 and 5)

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

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DSGB wrote:
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

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...

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