Question 378532
Now {{{60 = 2^2 * 3*5}}}.  Since {{{12 = 2^2 * 3}}} is supposed to be the GCF of n and 60, n must be of the form {{{n = 2^2 * 3 *p}}}, 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.