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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?

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.

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 _________________

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 (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;

Re: The “connection” between any two positive integers a and b [#permalink]

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23 Jun 2015, 15:04

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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 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.

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.

Re: The “connection” between any two positive integers a and b [#permalink]

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14 Apr 2016, 01:53

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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 ?
_________________

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.

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..

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