GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 14 Oct 2019, 03:43 GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  question on number of factors

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Director  Status: Done with formalities.. and back..
Joined: 15 Sep 2012
Posts: 563
Location: India
Concentration: Strategy, General Management
Schools: Olin - Wash U - Class of 2015
WE: Information Technology (Computer Software)
question on number of factors  [#permalink]

Show Tags

1
Hi,

I've a question on finding out number of even and odd factors of a number. Is there any quick method to identify how many 'even' or 'odd' factors a number would have.
Eg. how many 'even' factors does 144n have, where n is a prime number greater than 2. we can quickly identify, since 144=2^4*3^2 so there would be 30 factors of 144n but, how many even and how many odd?

Thanks.
_________________
Lets Kudos!!! Black Friday Debrief
Manager  Joined: 27 Feb 2012
Posts: 113
Re: question on number of factors  [#permalink]

Show Tags

Vips0000 wrote:
Hi,

I've a question on finding out number of even and odd factors of a number. Is there any quick method to identify how many 'even' or 'odd' factors a number would have.
Eg. how many 'even' factors does 144n have, where n is a prime number greater than 2. we can quickly identify, since 144=2^4*3^2 so there would be 30 factors of 144n but, how many even and how many odd?

Thanks.

Total Number of factors: (4+1)(2+1)(2) = 30
odd Factors Only = 3*2 = 6
Even facors = 30-6=24

Check for 12
factors 1,2,3,4,6,12
Total factors = 6
Even = 6-2=4
_________________
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Please +1 KUDO if my post helps. Thank you.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9697
Location: Pune, India
Re: question on number of factors  [#permalink]

Show Tags

Vips0000 wrote:
Hi,

I've a question on finding out number of even and odd factors of a number. Is there any quick method to identify how many 'even' or 'odd' factors a number would have.
Eg. how many 'even' factors does 144n have, where n is a prime number greater than 2. we can quickly identify, since 144=2^4*3^2 so there would be 30 factors of 144n but, how many even and how many odd?

Thanks.

You can arrive at the answer on your own if you understand (not just 'know' but 'understand') how to calculate the total number of factors of a number.

Say, a number N = 2^a * 3*b * 7^c

Total number of factors = (a + 1)(b + 1)(c + 1)

Why? because you can choose each prime number in (power + 1) ways, the additional 1 being for the case in which you don't choose the prime number. So you can select a 2 for the factor in (a + 1) ways.

You should go through this post first if you are not comfortable with this concept: http://www.veritasprep.com/blog/2010/12 ... ly-number/

Now, say the number is N = 2^a * 3*b * 7^c

How many factors will be even? What do you need for even factors? At least one 2. In how many ways can you do it?
In a*(b + 1)*(c + 1) ways
(the 1 additional way in which you don't pick a 2 has been removed. Rest everything is the same.)

How many will be odd? Now you don't want a 2. So, you get 1*(b+1)*(c +1 ) ways

Similarly, you can pose tons of questions: How many factors will be multiples of 4/6/21 etc? How many factors will have all the primes (2, 3 and 7)? etc
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Director  Status: Done with formalities.. and back..
Joined: 15 Sep 2012
Posts: 563
Location: India
Concentration: Strategy, General Management
Schools: Olin - Wash U - Class of 2015
WE: Information Technology (Computer Software)
Re: question on number of factors  [#permalink]

Show Tags

VeritasPrepKarishma wrote:
Vips0000 wrote:
Hi,

I've a question on finding out number of even and odd factors of a number. Is there any quick method to identify how many 'even' or 'odd' factors a number would have.
Eg. how many 'even' factors does 144n have, where n is a prime number greater than 2. we can quickly identify, since 144=2^4*3^2 so there would be 30 factors of 144n but, how many even and how many odd?

Thanks.

You can arrive at the answer on your own if you understand (not just 'know' but 'understand') how to calculate the total number of factors of a number.

Say, a number N = 2^a * 3*b * 7^c

Total number of factors = (a + 1)(b + 1)(c + 1)

Why? because you can choose each prime number in (power + 1) ways, the additional 1 being for the case in which you don't choose the prime number. So you can select a 2 for the factor in (a + 1) ways.

You should go through this post first if you are not comfortable with this concept: http://www.veritasprep.com/blog/2010/12 ... ly-number/

Now, say the number is N = 2^a * 3*b * 7^c

How many factors will be even? What do you need for even factors? At least one 2. In how many ways can you do it?
In a*(b + 1)*(c + 1) ways
(the 1 additional way in which you don't pick a 2 has been removed. Rest everything is the same.)

How many will be odd? Now you don't want a 2. So, you get 1*(b+1)*(c +1 ) ways

Similarly, you can pose tons of questions: How many factors will be multiples of 4/6/21 etc? How many factors will have all the primes (2, 3 and 7)? etc

Thanks, I got the quick method I was looking for.
_________________
Lets Kudos!!! Black Friday Debrief
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
Re: question on number of factors  [#permalink]

Show Tags

Vips0000 wrote:
Hi,

I've a question on finding out number of even and odd factors of a number. Is there any quick method to identify how many 'even' or 'odd' factors a number would have.
Eg. how many 'even' factors does 144n have, where n is a prime number greater than 2. we can quickly identify, since 144=2^4*3^2 so there would be 30 factors of 144n but, how many even and how many odd?

Thanks.

This thread may be old, but this doubt is still relevant. Hence reviving the discussion on this one. Vips' question is modeled on an easier Official question:

If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(This question is discussed in detail here)

I'll first explain the Official question briefly (to better establish the contrast with Vips' question later)

The first step to solving all such questions is to express the given number in its Prime Factorized form.

By doing that, we get: $$n=2^{2}*p^{1}$$

Had we been asked the total number of factors of n, the answer would have been: (2+1)(1+1) = 3*2 = 6 . (These 6 factors also include the cases of $$2^{0}*p$$ and 2$$^{0}*p^{0}$$)

But we are asked here, the number of even factors of n. Now, an even factor will always contain at least one 2. So, the cases in which the power of 2 is 0 are not allowed here. So, when asked about the number of even factors of n, we should only consider 2 powers of 2 ($$2^{1}$$ and $$2^{2}$$), not the 3 powers we considered in the question of total number of factors ($$2^{1}$$, $$2^{2}$$ and $$2^{0}$$)

So, the number of even factors of n = (2)(1+1) = 4

(Please refer to Karishma's lucid explanation above if you still have any doubt about the way we calculated the number of even factors )

Okay, so after having established this process of solving, let's now come to Vips' question:

How many 'even' factors does 144n have, where n is a prime number greater than 2?

Like before, our first step will be to do the prime factorization of the given number. $$144=2^{4}*3^{2}$$

So, $$144n=2^{4}*3^{2}*n$$

Now, n is a prime number greater than 2. Can n be 3? It very well can be!

So, 2 cases will arise here (this is the point where Vips' question differs from the Official question above)

I) If n = 3

Then $$144n=2^{4}*3^{3}$$

So, by applying the same logic as in the Official question above, the total number of even factors of 144n = (4)(3+1) = 16

II) If n is a prime greater than 3

Then, $$144n=2^{4}*3^{2}*n^{1}$$
So, the total number of even factors of 144n = (4)(2+1)(1+1) = 24

So, we see that in the question posed by Vips', the number of even factors of 144n may be either 16 or 24. A unique answer cannot be determined (because n may be 3 or greater than 3)

Takeaways:

1. Pay attention to the condition given in question (the question mentioned only that n was a prime greater than 2. So, the case of n = 3 did need to be considered as well)

2. Don't do calculations in the air (After prime factorizing 144, had Vips' written the next step: $$144n=2^{4}*3^{2}*n$$ on paper, the possibility of n = 3 may have occurred to him. However, when we do things in the air, since our mind is already occupied by trying to remember all the numbers that are being multiplied and process the multiplication at the same time, the likelihood of us noticing something amiss decreases significantly)

Hope this discussion was helpful! - Japinder
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 13088
Re: question on number of factors  [#permalink]

Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: question on number of factors   [#permalink] 28 Jan 2018, 23:59
Display posts from previous: Sort by

question on number of factors

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  