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 Your Progress

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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

For all positive integers f, f◎ equals the distinct pairs of [#permalink]

Show Tags

15 Nov 2010, 13:34

1

This post received KUDOS

4

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

55% (hard)

Question Stats:

69% (02:53) correct
31% (01:50) wrong based on 124 sessions

HideShow timer Statistics

For all positive integers f, f◎ equals the distinct pairs of positive integer factors. For example, 16◎ =3, since there are three positive integer factor pairs in 16: 1 x 16, 2 x 8, and 4 x 4.

What is the greatest possible value for f◎ if f is less than 100?

Great question - and my strategy for something like this is to first look at what they're asking:

The greatest number...

Which means that I want to maximize the number of factors, meaning that I also want to minimize the value of each factor (so that I can use it more often):

For example, if I use 99 (9 * 11), that 11 takes up too much space...I can't use it very often. Whereas if I use smaller factors (2 and 3), I can use them more often. 2 can become 4 and then 8 before it takes up as much space as does 11.

So if my goal is to find a number less than 100 that has as many small factors as possible, I'm looking at 96, because:

99 = 3*3*11 (and 11 is too big a prime factor) 98 = 2*7*7 (and 7 is too big a prime factor) 97 = not divisible by 2 or 3, so you're not using your smallest available factors 96 = 2*2*2*2*2*3 (or 32*3) ---> this is perfect because you're getting maximum value out of your factors. The only other that would work is 64 (drop the 3 and add another 2)

So for 96, break out the factors into pairs:

1, 96 2, 48 4, 24 8, 12 (these are easy to do - just double one number and halve the other for a fresh pair) 16, 6 32, 3

And that's as far as you can go. My fear is that we may not have enough unique factors (there's a lot of repetitiveness with the 2s), so I'll check that by trying the next smallest prime factor, 5. The biggest I can go with that is 90, or 2*3*3*5, and that gets us:

1, 90 2, 45 3, 30 5, 18 6, 15 9, 10

We're still at 6 factors, and it's going to get harder and harder to incorporate larger prime factors and have room for multiple factors in between (we've seen that 7 and 11 limit us a lot), so the answer must be 6. _________________

For all positive integers f, f◎ equals the distinct pairs of positive integer factors. For example, 16◎ =3, since there are three positive integer factor pairs in 16: 1 x 16, 2 x 8, and 4 x 4.

What is the greatest possible value for f◎ if f is less than 100? a)6 b)7 c)8 d)9 e)10

As stated earlier we need to maximize the number of factors. This can be done by using the smallest possible base and the highest possible power. 1: 2^6 = 64 => (1,2,4,8;8,16,32,64) = > this gives us 4 pairs. Though this need not give us the answer it gives us the highest power => 6. So any subsequent answer would have the sum of powers not more than 6. 2: 2^5 * 3 = 96 = > (1,2,3,4,6,8;12,16,24,32,48,96) = > this gives us 6 pairs Other combinations such as 2^4*3^2, 2^5*5, etc would be more than 100. Answer: A-6

Re: For all positive integers f, f◎ equals the distinct pairs of [#permalink]

Show Tags

05 Feb 2014, 12:55

tejhpamarthi wrote:

For all positive integers f, f◎ equals the distinct pairs of positive integer factors. For example, 16◎ =3, since there are three positive integer factor pairs in 16: 1 x 16, 2 x 8, and 4 x 4.

What is the greatest possible value for f◎ if f is less than 100?

A. 6 B. 7 C. 8 D. 9 E. 10

Brian, can this be thought in the following way?

When one does prime factorization you get one factor pair for every prime number (counting repetitions). So we are basically asked how many primes can we have in this factorization so that x<100.

Well I start with 100, and 2 being the smallest prime I can get and not until 2^6 do I get a number that is smaller than 100. So that's why I chose A

Re: For all positive integers f, f◎ equals the distinct pairs of [#permalink]

Show Tags

18 Jul 2015, 22:02

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: For all positive integers f, f◎ equals the distinct pairs of [#permalink]

Show Tags

19 Nov 2015, 05:38

We can also solve this by calculating from the answer option backwards and a simple P&C formula we use to calculate the number of factors, though I was not able to think through the entire thing in the first go

Formula - If a number N can be written as a product like this: P^a * Q^b.... where P,Q etc. are prime numbers and a,b...are the highest powers of these primes in the numbers, then the total number of factors for number N is given by (a+1)(b+1)(c+1) <One more additional fact here is that you'll get odd # factors for perfect squares only>

So now since we need to find pairs of factors in our question --> Let us say we have a total of N factors. Then we need to select 1 number out of N/2 factors i.e. N/2 C 1 to identify the number of pairs (since if we select one from the half, the other from the remaining half will be a fix selection)

So now back calculating from our options: A. 6 i.e total of 12 factors which is the maximum possible under 100 i.e. for 96 All others simply get eliminated automatically

I hope this makes sense

gmatclubot

Re: For all positive integers f, f◎ equals the distinct pairs of
[#permalink]
19 Nov 2015, 05:38

This is the kickoff for my 2016-2017 application season. After a summer of introspect and debate I have decided to relaunch my b-school application journey. Why would anyone want...

Check out this awesome article about Anderson on Poets Quants, http://poetsandquants.com/2015/01/02/uclas-anderson-school-morphs-into-a-friendly-tech-hub/ . Anderson is a great place! Sorry for the lack of updates recently. I...

Time is a weird concept. It can stretch for seemingly forever (like when you are watching the “Time to destination” clock mid-flight) and it can compress and...