# How many divisors does the product have?

• Jun 19th 2011, 10:28 PM
skyd171
How many divisors does the product have?
If p and q are prime numbers, how many divisors does the product $\displaystyle p^3*q^6$ have?

Prime factorization would suggest that the answer is 18, but apparently it is 28. Please explain. Thanks.
• Jun 19th 2011, 10:34 PM
Drexel28
Re: How many divisors does the product have?
Quote:

Originally Posted by skyd171
If p and q are prime numbers, how many divisors does the product $\displaystyle p^3*q^6$ have?

Prime factorization would suggest that the answer is 18, but apparently it is 28. Please explain. Thanks.

You forgot the cases $\displaystyle 1,p,\cdots,p^3,q,\cdots,q^6$. In general $\displaystyle \sigma_0(p^nq^m)=(n+1)(m+1)$, recall that $\displaystyle \sigma_0$ is multiplicative and $\displaystyle \sigma_0(p^n)=n+1$.
• Jun 19th 2011, 10:41 PM
skyd171
Re: How many divisors does the product have?
wonderful. Is it correct that the sigma, in this context, means "the number of unique divisors of"?
• Jun 19th 2011, 10:43 PM
Drexel28
Re: How many divisors does the product have?
Quote:

Originally Posted by skyd171
wonderful. Is it correct that the sigma, in this context, means "the number of unique divisors of"?

I don't know what unique divisor means, but it is the number of divisors, it is a special case of the more general divisor function.