# 125 and 512; 256 and 625

For how many positive integers $$n<1000$$, do there exist integers $$a,$$ $$b,$$ $$c,$$ and $$d,$$ that satisfy the following conditions:

$\begin{cases} 100a+b =c^n\\ 10b+a=d^n\\ gcd(c,d)=1? \end{cases}$

Details and assumptions

There are no restrictions on the values of $$a$$ and $$b$$.

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