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.