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The “connection” between any two positive integers a and b [#permalink]
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29 Feb 2012, 08:24
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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
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 Aravind Chembeti



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Re: Problem related to LCM [#permalink]
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29 Feb 2012, 09:12
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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 coprime (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 coprime 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 (182)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive; There are (183)/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+63=12; Total integers in the range from 0 to 20, not inclusive is 19; Hence, there are total of 1912=7 numbers which have no common factor with 6 other than 1: 1, 5, 7, 11, 13, 17 and 19. Answer: A.
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Re: The “connection” between any two positive integers a and b [#permalink]
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21 Sep 2012, 09:54
Hi, Can you please explain what method have you used below:
There are (182)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive; There are (183)/3+1=6 multiples of 3 in the range from 0 to 20, not inclusive;
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Re: The “connection” between any two positive integers a and b [#permalink]
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22 Sep 2012, 02:10
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.
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Re: The “connection” between any two positive integers a and b [#permalink]
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Re: The “connection” between any two positive integers a and b [#permalink]
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27 Jun 2015, 09:22
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 coprime (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 coprime 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 (182)/2+1=9 multiples of 2 in the range from 0 to 20, not inclusive; There are (183)/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+63=12; Total integers in the range from 0 to 20, not inclusive is 19; Hence, there are total of 1912=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 noninclusive numbers is? After all, it seems if the you know that (6,y) are coprime, counting the number of primes excluding 2 or 3 would lead to the answer quicker.



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Re: The “connection” between any two positive integers a and b [#permalink]
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22 Dec 2015, 11:25
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.
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Re: The “connection” between any two positive integers a and b [#permalink]
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05 Feb 2016, 18:10
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.



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Re: The “connection” between any two positive integers a and b [#permalink]
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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 ?
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Re: The “connection” between any two positive integers a and b [#permalink]
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14 Apr 2016, 02:45
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 COPRIMES 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 = 1996+3 = 7
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Re: The “connection” between any two positive integers a and b [#permalink]
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14 Apr 2016, 04:34
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.



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Re: The “connection” between any two positive integers a and b [#permalink]
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14 Apr 2016, 06:59
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
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Re: The “connection” between any two positive integers a and b [#permalink]
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07 Aug 2017, 21:36
Can you please elaborate on this concept and thanks in advance so Actually we are looking for COPRIMES 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 = 1996+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 COPRIMES 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 = 1996+3 = 7




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