The “connection” between any two positive integers

The positive integer

*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_{}The positive integer

*y*is less than 20 and the connection between*y*and 6 is equal to_{}. How many possible values of*y*are there?7 | |

8 | |

9 | |

10 | |

11 |

Answer Explantion:

If the connection between

*y*and 6 is_{}, then the least common multiple of*y*and 6 must equal the product 6*y*. The least common multiple of two numbers equals the product of the two numbers only when there are no common factors (other than 1). Since*y*is a positive integer less than 20, check all the integers from 1 to 19 to see which ones have no factors greater than 1 in common with 6: 1, 5, 7, 11, 13, 17, and 19. So there are 7 possible values for*y*. -------------------------------------------------------------------

**In my own words**(which I can actually understand!),

for the ratio to be 1:1, the LCM should be =6y (so that it cancels out with the product).

For this to be possible, y should NOT be a multiple of 6, and should not include 2,3

I actually randomly guessed on this one cos I knew it would take me more thinking to get the right answer, and so I tried applying the pacing advice I read on a MGMAT forum today.

**or any multiples of 2,3.**The only numbers that are not multiples of 2,3 and 6 are 1,5,7,11,13,17,19, a total of 7 numbers!**Why I got it wrong:**I actually randomly guessed on this one cos I knew it would take me more thinking to get the right answer, and so I tried applying the pacing advice I read on a MGMAT forum today.

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