NEW HERE? LEARN MORE
closeclose
Last visit was: 26 Apr 2024, 18:40 It is currently 26 Apr 2024, 18:40
Decision Tracker
My Rewards
New posts
New comers' posts

Close
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
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

The “connection” between any two positive integers a and b

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
Chembeti
Manager
Manager
Joined: 25 Nov 2011
Posts: 126
Own Kudos [?]: 870 [56]
Given Kudos: 20
Location: India
Concentration: Technology, General Management
GPA: 3.95
WE:Information Technology (Computer Software)
Send PM
The “connection” between any two positive integers a and b [#permalink] New post   29 Feb 2012, 09:24
6
Kudos
48
Bookmarks
00:00
A
B
C
D
E

Difficulty:

55% (hard)

Question Stats:

67% (02:48) correct 33% (02:50) wrong based on 668 sessions
Hide Show timer Statistics
The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

A. 7
B. 8
C. 9
D. 10
E. 11
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92948
Own Kudos [?]: 619247 [14]
Given Kudos: 81609
Send PM
CAT Tests Forum Quiz Mode
Re: Problem related to LCM [#permalink] New post   29 Feb 2012, 10:12
6
Kudos
8
Bookmarks
Expert Reply
Chembeti wrote:
The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

A. 7
B. 8
C. 9
D. 10
E. 11


Since “connection” between y and 6 is 1/1 then LCM(6, y)=6y, which means that 6 and y are co-prime (they do not share any common factor but 1), because if the had any common factor but 1 then LCM(6, y) would be less than 6y.

So, we should check how many integers less than 20 are co-prime with 6, which can be rephrased as how many integers less than 20 are not divisible by 2 or 3 (6=2*3).

There are (18-2)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive;
There are (18-3)/3+1=6 multiples of 3 in the range from 0 to 20, not inclusive;
There are 3 multiples of 6 in the range from 0 to 20, not inclusive (6, 12, 18) - overlap of the above two sets;
Total multiples of 2 or 6 in the range from 0 to 20, not inclusive is 9+6-3=12;

Total integers in the range from 0 to 20, not inclusive is 19;

Hence, there are total of 19-12=7 numbers which have no common factor with 6 other than 1: 1, 5, 7, 11, 13, 17 and 19.

Answer: A.
General Discussion
avatar
154238
Intern
Intern
Joined: 09 Sep 2012
Posts: 26
Own Kudos [?]: 200 [0]
Given Kudos: 33
Location: United States
Send PM
Re: The “connection” between any two positive integers a and b [#permalink] New post   21 Sep 2012, 10:54
Hi,
Can you please explain what method have you used below:

There are (18-2)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive;
There are (18-3)/3+1=6 multiples of 3 in the range from 0 to 20, not inclusive;


Thanks
User avatar
sanjoo
Senior Manager
Senior Manager
Joined: 06 Aug 2011
Posts: 269
Own Kudos [?]: 596 [0]
Given Kudos: 82
Send PM
Re: The “connection” between any two positive integers a and b [#permalink] New post   22 Sep 2012, 03:10
I took apprx 3 to 3.5 min to solve this question..

i tried every posibility from 1 to 7 then i get at that point , what bunuel saying ...num should not b multiple of 2 and 3...

1,5,7,11,13,17,19... all primes less than 20 except 2 and 3.
User avatar
mejia401
Senior Manager
Senior Manager
Joined: 15 Sep 2011
Posts: 258
Own Kudos [?]: 1371 [0]
Given Kudos: 46
Location: United States
WE:Corporate Finance (Manufacturing)
Send PM
GMAT ToolKit User
Re: The “connection” between any two positive integers a and b [#permalink] New post   27 Jun 2015, 10:22
Bunuel wrote:
Chembeti wrote:
The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

A. 7
B. 8
C. 9
D. 10
E. 11


Since “connection” between y and 6 is 1/1 then LCM(6, y)=6y, which means that 6 and y are co-prime (they do not share any common factor but 1), because if the had any common factor but 1 then LCM(6, y) would be less than 6y.

So, we should check how many integers less than 20 are co-prime with 6, which can be rephrased as how many integers less than 20 are not divisible by 2 or 3 (6=2*3).

There are (18-2)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive;
There are (18-3)/3+1=6 multiples of 3 in the range from 0 to 20, not inclusive;
There are 3 multiples of 6 in the range from 0 to 20, not inclusive (6, 12, 18) - overlap of the above two sets;
Total multiples of 2 or 6 in the range from 0 to 20, not inclusive is 9+6-3=12;

Total integers in the range from 0 to 20, not inclusive is 19;

Hence, there are total of 19-12=7 numbers which have no common factor with 6 other than 1: 1, 5, 7, 11, 13, 17 and 19.

Answer: A.


Hello,

Could you or someone else elaborate what the value of counting non-inclusive numbers is? After all, it seems if the you know that (6,y) are co-prime, counting the number of primes excluding 2 or 3 would lead to the answer quicker.
User avatar
G
Nevernevergiveup
Retired Moderator
Joined: 18 Sep 2014
Posts: 1015
Own Kudos [?]: 2755 [3]
Given Kudos: 79
Location: India
Send PM
GMAT ToolKit User Reviews Badge
Re: The “connection” between any two positive integers a and b [#permalink] New post   22 Dec 2015, 12:25
3
Kudos
The “connection” between any two positive integers a and b is the ratio of

  • the smallest common multiple of a and b ...........i.e., LCM(least common multiple)
  • the product of a and b.


For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

the ratio is 1:1 i.,e., LCM and product are equal.

this can happen only when there is no common factor(other than 1) between y and 6.

since 6 has two factors 2 and 3.

y can have values as all those numbers which are not multiples of 2 and 3 and less than 20.

y={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}

y={1,5,7,11,13,17,19}

so total 7 values are possible for y..................A is the answer.
User avatar
P
mvictor
Board of Directors
Joined: 17 Jul 2014
Posts: 2163
Own Kudos [?]: 1180 [0]
Given Kudos: 236
Location: United States (IL)
Concentration: Finance, Economics
Schools: Stanford '20 (WD)
GMAT 1: 650 Q49 V30
GPA: 3.92
WE:General Management (Transportation)
Send PM
GMAT ToolKit User Reviews Badge
Re: The “connection” between any two positive integers a and b [#permalink] New post   05 Feb 2016, 19:10
it can be 1/1 only if y is a prime number or 1, otherwise, it will be messed up. 3 won't work here, because LCM of 3 and 6 is 6, and 6/3*6 is not 1/1.
thus, we are left with 1, 5, 7, 11, 13, 17, and 19.
7 options.
User avatar
D
stonecold
Alum
Joined: 12 Aug 2015
Posts: 2282
Own Kudos [?]: 3132 [4]
Given Kudos: 893
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Send PM
GMAT ToolKit User
Re: The “connection” between any two positive integers a and b [#permalink] New post   14 Apr 2016, 02:53
3
Kudos
1
Bookmarks
Hey chetan2u can you look at my solution for this one=>

Here connection = LCM / product = 1/HCF
now Connection between Y and ^=1/1
so the HCF must be 1
so the values possible are => 1,5,7,11,13,17,19
So 7 values
Hence A
Am i missing something here ?
User avatar
L
chetan2u
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11181
Own Kudos [?]: 31969 [2]
Given Kudos: 291
Send PM
Reviews Badge
Re: The “connection” between any two positive integers a and b [#permalink] New post   14 Apr 2016, 03:45
1
Kudos
1
Bookmarks
Expert Reply
stonecold wrote:
Hey chetan2u can you look at my solution for this one=>

Here connection = LCM / product = 1/HCF
now Connection between Y and ^=1/1
so the HCF must be 1
so the values possible are => 1,5,7,11,13,17,19
So 7 values
Hence A
Am i missing something here ?


Hi,
you are absolutely correct with the logic and concept behind this Q..
HCF * LCM = product of two numbers..
so IF 'connection' is 1, LCM/(LCM*HCF) is 1 or HCF = 1, as correctly pointed by you..

so Actually we are looking for CO-PRIMES to 6..
factors of 6 are 2 and 3..
in first 19 digits 19/2 or 9 are multiples of 2..
19/3 or 6 are multiple of 3, out of which 19/6 or 3 are already catered for in multiples of 2 above ..
SO total = 19-9-6+3 = 7
User avatar
HarisinghKhedar
Intern
Intern
Joined: 21 Oct 2015
Posts: 38
Own Kudos [?]: 13 [0]
Given Kudos: 115
GMAT 1: 620 Q47 V28
Send PM
GMAT ToolKit User
Re: The “connection” between any two positive integers a and b [#permalink] New post   14 Apr 2016, 05:34
Do you guys think that 6 would make a good trap answer here?

I just started this problem by factoring 6 = 2*3. With additional clue that Y should not have these two factors I went on to evaluate subsequent numbers <20 and ended up with 6 numbers. And then, I realized that I didn't count 1.
User avatar
D
stonecold
Alum
Joined: 12 Aug 2015
Posts: 2282
Own Kudos [?]: 3132 [0]
Given Kudos: 893
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Send PM
GMAT ToolKit User
Re: The “connection” between any two positive integers a and b [#permalink] New post   14 Apr 2016, 07:59
HarisinghKhedar wrote:
Do you guys think that 6 would make a good trap answer here?

I just started this problem by factoring 6 = 2*3. With additional clue that Y should not have these two factors I went on to evaluate subsequent numbers <20 and ended up with 6 numbers. And then, I realized that I didn't count 1.



Maybe ..
P.S => Never ever forget zero or 1
they are core to the gmat..

regards
Stone Cold
User avatar
B
gps5441
Manager
Manager
Joined: 04 May 2014
Posts: 114
Own Kudos [?]: 72 [0]
Given Kudos: 126
Location: India
WE:Sales (Mutual Funds and Brokerage)
Send PM
Re: The “connection” between any two positive integers a and b [#permalink] New post   07 Aug 2017, 22:36
Can you please elaborate on this concept and thanks in advance

so Actually we are looking for CO-PRIMES to 6..
factors of 6 are 2 and 3..
in first 19 digits 19/2 or 9 are multiples of 2..
19/3 or 6 are multiple of 3, out of which 19/6 or 3 are already catered for in multiples of 2 above ..
SO total = 19-9-6+3 = 7[/quote



chetan2u wrote:
stonecold wrote:
Hey chetan2u can you look at my solution for this one=>

Here connection = LCM / product = 1/HCF
now Connection between Y and ^=1/1
so the HCF must be 1
so the values possible are => 1,5,7,11,13,17,19
So 7 values
Hence A
Am i missing something here ?


Hi,
you are absolutely correct with the logic and concept behind this Q..
HCF * LCM = product of two numbers..
so IF 'connection' is 1, LCM/(LCM*HCF) is 1 or HCF = 1, as correctly pointed by you..

so Actually we are looking for CO-PRIMES to 6..
factors of 6 are 2 and 3..
in first 19 digits 19/2 or 9 are multiples of 2..
19/3 or 6 are multiple of 3, out of which 19/6 or 3 are already catered for in multiples of 2 above ..
SO total = 19-9-6+3 = 7
User avatar
G
testcracker
Manager
Manager
Joined: 24 Mar 2015
Status:love the club...
Posts: 220
Own Kudos [?]: 112 [0]
Given Kudos: 527
Send PM
The “connection” between any two positive integers a and b [#permalink] New post   15 Mar 2018, 19:02
Chembeti wrote:
The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1. How many possible values of y are there?

A. 7
B. 8
C. 9
D. 10
E. 11



there are a total of 8 primes (2, 3 ,5 ,7, 11, 13, 17, 19) between 1 and 20 inclusive, and out of these 8 primes, y can take on 6 values, because y cannot be 2 or 3

so there exist 6 values plus 1 = 7 possible values of y

thanks
:cool:
User avatar
L
Kinshook
GMAT Club Legend
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5344
Own Kudos [?]: 3968 [0]
Given Kudos: 160
Location: India
Schools: HBS '22 CBS '22 Wharton '22
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Send PM
Reviews Badge
The connection between any two positive integers a and b [#permalink] New post   28 Sep 2022, 20:19
Given: The “connection” between any two positive integers a and b is the ratio of the smallest common multiple of a and b to the product of a and b. For instance, the smallest common multiple of 8 and 12 is 24, and the product of 8 and 12 is 96, so the connection between 8 and 12 is 24/96 = 1/4

The positive integer y is less than 20 and the connection between y and 6 is equal to 1/1.

Asked: How many possible values of y are there?

Connection (a, b) = LCM(a, b)/ab = 1/ HCF(a, b)

Connection (y, 6) = 1 = 1/HCF(y, 6)
There is no common factor between y & 6 and y is less than 20

y = {1,5,7,11,13,17,19}: 7 values

IMO A
User avatar
S
samarpan.g28
Manager
Manager
Joined: 08 Dec 2023
Posts: 206
Own Kudos [?]: 29 [0]
Given Kudos: 1165
Location: India
Concentration: Strategy, Operations
Schools: HEC Montreal ISB IIM Rotman HEC INSEAD Jan'25 intake
GPA: 4
WE:Engineering (Tech)
Send PM
Re: The connection between any two positive integers a and b [#permalink] New post   17 Dec 2023, 10:05
LCM=6Y.
Given, Y<20. This is applicable for prime nos less than 20 (excluding 2 and 3), and of course 1. So, 1,5,7,11,13,17,19 -> 7 (A).
GMAT Club Bot
gmatclubot
Re: The connection between any two positive integers a and b [#permalink]
17 Dec 2023, 10:05
Moderators:
Bunuel
Math Expert
92948 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions