Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 22 Sep 2010
Posts: 9

For all positive integers f, f◎ equals the distinct pairs of [#permalink]
Show Tags
15 Nov 2010, 13:34
1
This post received KUDOS
5
This post was BOOKMARKED
Question Stats:
70% (02:51) correct
30% (01:58) wrong based on 162 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? A. 6 B. 7 C. 8 D. 9 E. 10
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
tejh



Veritas Prep GMAT Instructor
Joined: 26 Jul 2010
Posts: 244

Re: problem solving [#permalink]
Show Tags
15 Nov 2010, 14:17
2
This post received KUDOS
1
This post was BOOKMARKED
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.
_________________
Brian
Save $100 on live Veritas Prep GMAT Courses and Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.
Veritas Prep Reviews



Manager
Joined: 03 Jun 2010
Posts: 181
Location: United States (MI)
Concentration: Marketing, General Management
WE: Business Development (Consumer Products)

Re: problem solving [#permalink]
Show Tags
16 Nov 2010, 09:19
Used the same logic, but worked with 80. What's the OA?
Posted from my mobile device



Director
Status: =Given to Fly=
Joined: 04 Jan 2011
Posts: 834
Location: India
Concentration: Leadership, Strategy
GMAT 1: 650 Q44 V37 GMAT 2: 710 Q48 V40 GMAT 3: 750 Q51 V40
GPA: 3.5
WE: Education (Education)

Re: Try this one! [#permalink]
Show Tags
05 Feb 2011, 21:17
2
This post received KUDOS
Is it 96? With 6 pairs? Method: Maximum power of 2 < 100 is 2^6=64 But we need to create distinct pairs right? so I removed a two and put in a 3 instead :D 2^5x3=96
_________________
"Wherever you go, go with all your heart"  Confucius
Useful Threads
1. How to Review and Analyze your Mistakes (Post by BB at GMAT Club)
2. 4 Steps to Get the Most out out of your CATs (Manhattan GMAT Blog)
My Experience With GMAT
1. From 650 to 710 to 750  My Tryst With GMAT
2. Quest to do my Best  My GMAT Journey Log



Math Forum Moderator
Joined: 20 Dec 2010
Posts: 2010

Re: Try this one! [#permalink]
Show Tags
05 Feb 2011, 23:09
What's the idea; I took numbers in reverse order from 99 and got 6 distinct pairs in 96. I thought it may be because 96=(2^5*3) has (5+1)*(1+1)=12(perhaps maximum) factors. 1*96,2*48*4*24,8*12,16*6,32*3 However, I was not sure while answering this. Please let us know if there is a better way. Ans: "A"
_________________
~fluke
GMAT Club Premium Membership  big benefits and savings



Current Student
Joined: 12 Jan 2012
Posts: 26

Re: problem solving [#permalink]
Show Tags
05 Jan 2013, 00:59
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 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: A6



Current Student
Joined: 06 Sep 2013
Posts: 1998
Concentration: Finance

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 Is this method correct or is it rather flawed? Thanks Cheers J



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15919

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.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



Intern
Joined: 28 Nov 2012
Posts: 39

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



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15919

Re: For all positive integers f, f◎ equals the distinct pairs of [#permalink]
Show Tags
17 Feb 2017, 07:33
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.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: For all positive integers f, f◎ equals the distinct pairs of
[#permalink]
17 Feb 2017, 07:33







