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

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    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.

    Would the answer just be

    (p^m-m^(m-1))(q^n-q^(n-1))?
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

    Quote Originally Posted by ehpoc View Post
    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?
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    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?
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

    Quote Originally Posted by ehpoc View Post
    I am not concerned.
    Pray tell what does that mean?

    Quote Originally Posted by ehpoc View Post
    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.

    If you read reply #2 carefully it contains the answer to the OP.
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

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

    Would the answer just be

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

    Find the number of distinct divisors of 2^23^3
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    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.
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    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.
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

    Ya I was never thinking about this question properly.

    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?
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

    Quote Originally Posted by ehpoc View Post
    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.
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    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)?
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    Re: If p and q are distinct primes, find the number of distinct divisors of (p^m)(q^n

    Yes.
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