How many divisors does the product have?

**skyd171** · June 19th 2011, 10:28 PM

If p and q are prime numbers, how many divisors does the product $p^3*q^6$ have?

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

**Drexel28** · June 19th 2011, 10:34 PM

Originally Posted by skyd171

If p and q are prime numbers, how many divisors does the product $p^3*q^6$ have?

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

You forgot the cases $1,p,\cdots,p^3,q,\cdots,q^6$ . In general $\sigma_0(p^nq^m)=(n+1)(m+1)$ , recall that $\sigma_0$ is multiplicative and $\sigma_0(p^n)=n+1$ .

**skyd171** · June 19th 2011, 10:41 PM

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

**Drexel28** · June 19th 2011, 10:43 PM

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.

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