Find the smallest integer greater than 1 that has a square root, a cube root, a fourth root, a fifth root, a sixth root, a seventh root, an eighth root, a ninth root, and a tenth root, all perfect.
where x = lcm(2,3,4,...,10).
By the way, where did you get these problems?
EDIT: I might have misinterpreted this problem. Thinking of perfect powers, I interpreted that the square root through tenth root are all integer. But if we further require that they be perfect numbers then I have no idea what the solution looks like.
Clearly for each we must have is divisible by . Hence the smallest natural with the given property is:
because this number has the required property and any other number with this property must be the product of powers of primes each such power also having the property etc.