How many positive divisors are there of 30^2024 which are not divisors of 20^2021?
I have tried many ways to try to get a pattern for this problem but I can't. I know that 30 has 8 divisors and 20 has 6 divisors, and the number of divisors in 30 not present in 20 is 4, But I just can't proceed from there.
Use the prime factorization of the given numbers, i.e. the fact that any integer number $x \in \mathbb{N}$ has a unique factorization $x=p_1^{r_1}\cdot\dots\cdot p_k^{r_k}$ with $p_1 < \dots < p_k$ prime numbers and $r_i \in \mathbb{N}^*$. The divisors of $x$ are then just all numbers of the form $p_1^{s_1} \cdot \dots \cdot p_k^{s_k}$ with $s_i \in \{0,1,\dots,r_i\}$. Therefore, the divisors of $x^n$ are all numbers of the form $p_1^{s_1} \cdot \dots \cdot p_k^{s_k}$ with $s_i \in \{0,1,\dots,nr_i\}$.
If you now apply this observation to the specific numbers given (and their prime factorization which you already found), it should be easy to see which of the divisors of $30^{2024}$ are also divisors of $20^{2021}$ (and then count them).