125 and 512; 256 and 625

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

{100a+b=cn10b+a=dngcd(c,d)=1?\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 aa and bb.

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