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 followingresult: Let be the prime factorization of . Then is a-thpower if and only if each exponent is divisible by .

; ; and .

We construct a system of congruence for .

Since , , and area fifth,a sixth, anda seventhpowers, respectively, theresultimplies 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 .

Interesting problem!