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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

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.

"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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"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

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.

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. Answer: B _________________

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.

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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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

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.

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

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17 Aug 2016, 03:12

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?

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 _________________

Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

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 _________________

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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Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

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

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

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26 Nov 2017, 01:38

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