# Show A_k={X is divisible by p_k} independent events

1. Feb 27, 2013

### Gregg

1. The problem statement, all variables and given/known data

$p_k$ refers to prime numbers $p_1<\cdots<p_k<\cdots$

I thought that it could have something to do with the FTA so all natural numbers are composed of a unique product of primes.

So I need to show $\text{P} ( X=M p_l \cap X = N p_m ) = \text{P} ( X=M p_l ) \text{P}( X = N p_m )$ and this shows that all of the events are independent doesn't it...

M,N are integers which are the product of the remaining primes. So actually, there is an implied union inside those brackets which means i have to take the union of all integers?