MHB How Many Divisors of $30^{2003}$ Are Not Divisors of $20^{2000}$?

[anemone]

Here is this week's POTW:

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How many divisors of the number $30^{2003}$ are not divisor of $20^{2000}$?

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[anemone]

Congratulations to kaliprasad for his correct solution, which you can find below:

Spoiler

Let us factorise $30^{2003}$ and $20^{2000}$

$30^{2003}= 3^{2003} * 2 ^{2003} * 5^{2003}$

as 2,3,5 are pairwise co-primes and in factor each can come 0 to 2003 times that is 2004 ways
so number of factors = $(2003+1) * (2003 + 1) * (2003 +1)= 2004^3$

Now
$20^{2000}= 2^{2000} * (2 * 5) ^{2000}= 2^ {4000} * 5^{2000}$

$\gcd (30^{2003},20^{2000}) = 2^{2003} * 5 ^{2000}$

Any number that divides $30^{2003}$ and $20^{2000}$ must divide $\gcd (30^{2003},20^{2000})$

So number of numbers that divide $30^{2003}$ and $20^{2000}$ = $(2003+1)(2000+1) = 2004 * 2001$

So number of numbers that divide $30^{2003}$ and does not divide $20^{2000}$ = $2004^3 - 2004 * 2001 = 2004(2004^2-2021) = 8044086060$

8044086060 is the number of divisors of $30^{2003}$ that are not divisor of $20^{2000}$.