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Since 678,463 is not divisible by 6. That means the only case where it could be a multiple of z is when k (in 6^k) is 0, as that would make your z value equal to 1, and we know that anything is divisible by 1.
For k to be equal to 0, our integer value which can range from 0-10 can only be 0 for 678,463 to be a mutiple of.
Since there are 5 numbers in the set, there can be 5 events where our z value is 0.
The total number of possible events is: (the size of our set)x(the number of available integer values) since there are 5 numbers in our set and 11 available integer values(0,1,2,3,4,5,6,7,8,9,10) the total number of possible events is : 11*5=55.
(probability of our number being a multiple of z) = (the number of events where our z value is 0)/(the total number of events) = 5/55 = .091
probablility of our number NOT being a multiple z) = 1 - (probability of our number being a multiple of z) = 1 - (.091) = 0.9
the answer is therefore 90%.