# If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n).

• December 19th 2011, 06:32 AM
ehpoc
If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n).
If p and q are distinct primes, find the number of distinct divisors of pmqn.

(p^m-m^(m-1))(q^n-q^(n-1))?
• December 19th 2011, 06:44 AM
Plato
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Quote:

Originally Posted by ehpoc
If p and q are distinct primes, find the number of distinct divisors of $p^mq^n$.

Any divisor of $p^mq^n$ has the form $p^jq^k$ where $0\le j\le m~\&~0\le k\le n~.$

How many does that make?
• December 19th 2011, 07:06 AM
ehpoc
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
I am not concerned. Let me rephrase the question.

I was told the formula for the number of distinct divisors of p^k is p^k-p^(k-1). Is this the correct formula?
• December 19th 2011, 07:20 AM
Plato
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Quote:

Originally Posted by ehpoc
I am not concerned.

Pray tell what does that mean?

Quote:

Originally Posted by ehpoc
I was told the formula for the number of distinct divisors of p^k is p^k-p^(k-1). Is this the correct formula?

No, that is not correct.
For example, there are ten distinct divisors of $2^9$.

• December 19th 2011, 08:13 AM
Also sprach Zarathustra
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Quote:

Originally Posted by ehpoc
If p and q are distinct primes, find the number of distinct divisors of pmqn.

(p^m-m^(m-1))(q^n-q^(n-1))?

Find the number of distinct divisors of $2^23^3$
• December 19th 2011, 08:31 AM
ehpoc
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Ok I think I know where I went wrong. I got the total number of relatively prime numbers to (p^m)(q^n)

(p^m)(q^n)-(p^m-m^(m-1))(q^n-q^(n-1))

Amount of numbers from 1 to (p^m)(q^n) subtract the amount of numbers that are relatively prime.
• December 19th 2011, 08:41 AM
emakarov
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Not relatively prime $\not\!\!\!\!\implies$ divisor.
• December 19th 2011, 09:00 AM
ehpoc
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

ok so the number of distinct divisors of p and q are 2 each if they are prime right? p and 1, and q and 1 respectively

So the number of distinct divisors of p^k is k+1?

2^3 has four distinct divisors 4 right? 1, 2, 2^2, 2^3

so (p^m)(q^n) has mn many divisors?
• December 19th 2011, 09:04 AM
emakarov
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Quote:

Originally Posted by ehpoc
so (p^m)(q^n) has mn many divisors?

There are m + 1 integers between 0 to m and n + 1 integers between 0 and n.
• December 19th 2011, 09:32 AM
ehpoc
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Sorry I kind of made a typo (I am becoming infamous for it unfortunately LOL)

(m+1)(n+1) many distinct divisors of (p^m)(q^n)?
• December 19th 2011, 09:34 AM
emakarov
Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n
Yes.