Question 378532: A positive interger,n, is picked at random between 1 and 500 (inclusive). Find the probability that the GCF of n and 60 is 12.
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Now  . Since  is supposed to be the GCF of n and 60, n must be of the form  , where p is relatively prime to 5. Those numbers are exactly the numbers NOT DIVISIBLE by 5.
Now 12 * 41 = 492 < 500, so we need to consider only numbers p that are relatively prime to 5 and less than 41. The numbers that are DIVISIBLE by 5 and less than 41 are 5, 10, 15,20,25,30,35,40. (The GCF for these is 60.)
Thus there are 41 - 8 = 33 numbers (less than 500) such that GCF(60, n) = 12. Therefore the probability is 33/500.
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