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

Question

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

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

Bunuel · Accepted Answer

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.

stonecold · Answer

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 ?

sanjoo · Answer

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.

Nevernevergiveup · Answer

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.

chetan2u · Answer

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

Dooperman · Answer

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.

Bunuel · Answer

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.

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

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