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

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Re: The “connection” between any two positive integers a and b [#permalink]
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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 ...........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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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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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
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Re: The connection between any two positive integers a and b [#permalink]
­Can't we assume y to be 6 ? Question does't mention numbers to be distinct.
6, 6 will also result in the ratio being 1.
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Re: The connection between any two positive integers a and b [#permalink]

Dooperman 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

­Can't we assume y to be 6 ? Question does't mention numbers to be distinct.
6, 6 will also result in the ratio being 1.

­
No, y cannot be 6. If y = 6, then the LCM of 6 and 6 is 6, and the product of 6 and 6 is 36. Hence, the “connection” between 6 and 6 is 6/36 = 1/6, not 1/1.­
Re: The connection between any two positive integers a and b [#permalink]
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