Exponent and Products of digits.

Question

The given digits, $2,3,4,5$ are to substitute inside the expression $$a^b \cdot c^d$$ to get $2020.2050,2075,2100,2016,2050$ .

Manually I've used trial and errors many times to get $24$ possible answers using the calculator. But the question does not allow the use of calculator. Is there anyone who knows how to solve it?

Answer 1

$2020$ is a multiple of $101$. $2016$ and $2100$ are multiples of $7$. $2050$ is a multiple of $41$. $2075$ is a multiple of $83$. None of these results are possible.

Answer 2

You can't get any of these numbers. If all of $a,b,c,d$ are $2,3,4$ or $5$, the only primes which can divide $a^bc^d$ are $2,3$ and $5$. All of the numbers in your list have other prime factors.