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

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The “connection” between any two positive integers a and b [#permalink] New post 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
[Reveal] Spoiler: OA

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Re: Problem related to LCM [#permalink] New post 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 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.
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Re: The “connection” between any two positive integers a and b [#permalink] New post 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;


Thanks
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Re: The “connection” between any two positive integers a and b [#permalink] New post 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] New post 23 Jun 2015, 15:04
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Re: The “connection” between any two positive integers a and b [#permalink] New post 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.
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Re: The “connection” between any two positive integers a and b [#permalink] New post 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] New post 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] 05 Feb 2016, 18:10
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