Relationships between powers of integers, Part 1

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Here is another question and (incomplete) answer from my Autumn 2020 FPM Piazza Forum. The question under discussion comes from my Sample Exam Paper, and the student was asking how to approach it, and whether the Fundamental Theorem of Arithmetic could help.

Here is the question:

sample question Question from FPM Sample Exam Paper, Autumn 2020

My (incomplete) answer:

I don't want to spoil this completely yet, though the solutions will be available soon!
You can (but don't have to) treat the easy special case where \(k=1\) first.
Can you do the special cases where \(k=2\) or \(k=10\)?
I don't think it does any harm to have the FTA in mind, though it may not help much with the proof. This question involves manipulation of powers of integers, and so I suggest you play around with powers of powers.
Best wishes,
Dr Feinstein

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