Ugly Powers

A Hamming Number (also called an ugly number) is any positive integer that has its set of prime factors as a subset of the prime numbers 2, 3, and 5. On the other hand, a prime power is produced using any integer $k$ raised to a prime $p$, and is represented as $p^k$.

As such, we can create a theoretical term, an “ugly power,” which is any ugly number multiplied by any arbitrary positive integer $k$. We can represent this by having any ugly number $h$, and an ugly power be $h^k$.

Write a function ugly_powers(s: set) -> bool which takes a set $s$ and returns a boolean value determining whether or not all the elements of set $s$ are all ugly powers.

Note: Can you write a solution that’s approximately $O(n\;log(n))$?

Example:

ugly_powers({1, 2, 5, 10}) -> True