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mahler1
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Homework Statement
Suppose one extracts a ball from a box containing ##n## numbered balls from ##1## to ##n##. For each ##1 \leq k \leq n##, we define ##A_k=\{\text{the number of the chosen ball is divisible by k}\}.##
Find ##P(A_k)## for each natural number which divides ##n##.
The Attempt at a Solution
I thought of thinking of ##P(A_k)## as ##P(A_k)=1-P({A_k}^c)##. And ##P({A_k}^c)=\{\text{the number of the chosen ball is not divisible by k}\}##. If ##k\geq 3##, then I know how many numbers less than ##k## are coprime with ##k## (Euler's totient function), however, in this case I would need the numbers greater than ##k## which are not divisible by ##k##. I don't know how to count them. I couldn't think of anything else, any advice or suggestions would be appreciated.
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