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prime factors

This topic has 3 expert replies and 3 member replies
aleph777 Master | Next Rank: 500 Posts Default Avatar
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prime factors

Post Thu Jan 27, 2011 12:34 pm
If n is a positive integer less than 200 and 14n⁄60 is also an integer, then n has how many different positive prime factors
a. 2
b. 3
c. 5
d. 6
e. 8

This is supposedly a GMAT PREP questions, and the OA is B

I'm curious about the phrasing, though. Just want to confirm that "different" means "unique" in this case?

When you solve for "unique" primes, then you end up with 2, 3, and 5, because the moment you multiply by 7 or greater, you violate the n < 200 restriction. But I was hesitant because I didn't know "different" meant "unique."

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Post Tue Dec 05, 2017 5:58 pm
aleph777 wrote:
If n is a positive integer less than 200 and 14n⁄60 is also 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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Post Wed Dec 06, 2017 10:44 am
Hi All,

We're told that N is a positive integer that is less than 200 and 14N⁄60 is also an integer. We're asked for the the number of different positive prime factors that N has. This question can be solved by TESTing VALUES. As long as we pick a value for N that fits the given 'restrictions', we'll get the correct answer.

IF.... N=60.... (14)(60)/(60) = 14 so we have a value that fits everything that we were told.

60 = (2)(2)(3)(5) so N has 3 different prime factors

Final Answer: B

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Post Fri Jan 28, 2011 6:41 am
B. good problem!!

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Post Fri Jan 28, 2011 12:24 am
aleph777 wrote:
If n is a positive integer less than 200 and 14n⁄60 is also an integer, then n has how many different positive prime factors
a. 2
b. 3
c. 5
d. 6
e. 8
14n/60 = 7n/30 is an integer.
Hence, n must be a multiple of 30 = 2*3*5

Thus, we know for sure that n has at least 3 different prime factors, i.e. 2, 3, and 5.

Now if n has one more different prime factor, let's say the minimum one, i.e. 7, minimum value of n will be 210 > 200. Hence n doesn't have any other prime factor.

The correct answer is B.

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aleph777 Master | Next Rank: 500 Posts Default Avatar
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Post Thu Jan 27, 2011 12:56 pm
I like this technique of yours:
14n/60 = 7n/30

Normally I'll just breakdown both numbers into primes, and then cut out any overlap to find n, but this way is way more elegant and much faster!

Thanks for the tip!

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Post Thu Jan 27, 2011 12:47 pm
aleph777 wrote:
If n is a positive integer less than 200 and 14n⁄60 is also an integer, then n has how many different positive prime factors
a. 2
b. 3
c. 5
d. 6
e. 8

This is supposedly a GMAT PREP questions, and the OA is B

I'm curious about the phrasing, though. Just want to confirm that "different" means "unique" in this case?

When you solve for "unique" primes, then you end up with 2, 3, and 5, because the moment you multiply by 7 or greater, you violate the n < 200 restriction. But I was hesitant because I didn't know "different" meant "unique."
different means unique here
0 step by step ...
14/60 --> 7/30
7n/30 --> n/30 as 0 n could be 6*30
prime fact-n of 180
180
--2--90
--2--45
--3--15
--3--5
--5--1
distinct prime factors 2,3,5 --> three prime factors

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