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Factors of product of n distinct primes 
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#1
Mar909, 06:48 PM

P: 74

What will be the numbers of positive factors of product of n distinct prime numbers?
i was able to get 2^n. pls how do i prove this? 


#2
Mar909, 07:02 PM

P: 287

In any factor you come up with, it will either be divisible by one of the n unique primes, or not. Whether or not your particular factor is divisible by a prime is independent of whether your factor is divisible by another prime; so how many ways can you make a factor if you have to make n yes/no decisions?
Think about it as a prime factor set. You're finding the cardinality of the power set... 


#3
Mar909, 07:09 PM

P: 74




#4
Mar909, 10:03 PM

P: 287

Factors of product of n distinct primes
Yes. Hence the 2^n.



#5
Mar1009, 03:03 AM

P: 74




#6
Mar1009, 06:19 AM

P: 287

You said "distinct primes" in your first post. So your answer  and my reason  were correct.
If you are now changing the question, that's fine. If the primes are not distinct, all you need to do is to calculate the number of equivalence classes of primes. For instance, P = {2, 2, 3, 3, 3, 5, 7, 7, 11, 11, 11, 11} => P' = { {2, 2}, {3, 3, 3}, {5}, {7, 7}, {11, 11, 11, 11} } Then the number of factors is 2^P' = 32. If the primes are all unique, then clearly P' = P. 


#7
Mar1009, 07:55 AM

Sci Advisor
HW Helper
P: 9,398

That is clearly incorrect: just consider a prime power. By your logic it always has exactly 2 divisors.
Google for Euler's phi (or totient) function. 


#8
Mar1009, 10:43 AM

P: 287

Oh, yeah. Sorry, I was still thinking in terms of the first question.
Matt Grime is correct for the second question. I was for the first one. Sorry for the confusion... 


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