Wednesday, November 4, 2009

CP #5

Question #2 from Kaplan Problem Solving challenge.
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 

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?





Answer Explantion:
If the connection between y and 6 is , then the least common multiple of y and 6 must equal the product 6y. 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 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.

No comments: