The hint suggests a sufficient form for that may satisfy the requirements of the problem.
Suppose that , where and are positive integers.
Now, I will apply the following result: Let be the prime factorization of . Then is a -th power if and only if each exponent is divisible by .
; ; and .
We construct a system of congruence for .
Since , , and are a fifth, a sixth, and a seventh powers, respectively, the result implies that , , and . Equivalently, we try to solve , , and ; thus, all solutions are given by . Check!
Similarly for , construct a system of three congruences and then solve it. I've found that . Check!
There are, in fact, infinitely many solutions for .
Example: Take and . Here . Indeed, ; ; and .
The smallest possible is obtained by taking and ; here .