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I hope the title is self-explanatory.

Source:

https://www.youtube.com/watch?v=lJ3CD9M3nEQ

time stamp 2:04

Bill Dubuque
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jxhyc
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  • I will always suggest checking the proof of uniqueness of prime factorization (part of the fundamental theorem of arithmetic). – Spectre Aug 12 '21 at 02:59
  • Or, just to hint it out, think: if a number has two different prime factorizations (here, take it to be the lowest number of its kind), couldn't you use Euclid's lemma to cancel out the prime factors in both the forms once you equate them? e.g. , I have a number $n$ of tqo different prime factorizations $\prod\limits_{i=1}^m p_i^{a_i}$ and $\prod\limits_{j=1}^n q_i^{b_i}$ ($p$,$q$ are primes). If I equate them both, can't I cancel each prime factor WLOG? – Spectre Aug 12 '21 at 03:01
  • Cancel here means dividing off from both sides... – Spectre Aug 12 '21 at 03:07

1 Answers1

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Let $n \in \mathbb{N}$ be a natural number such that $n = p_{1}^{\alpha_{1}}...p_{k}^{\alpha_{k}}$ = $q_{1}^{\beta_{1}}...q_{n}^{\beta_{n}}$ with $\{p_{1},...,p_{k},q_{1},...,q_{n}\}$ be distinct primes. What can you say about the divisor $q_{1}...q_{n}$?

Mike Desgrottes
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  • @MikeDesgorttes, speaking from my experience, I tell you not to answer simple questions like these as doing so should fetch you downvotes. I am already suffering for having done so. – Spectre Aug 12 '21 at 03:07
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    @Spectre noted, i will wait next time. – Mike Desgrottes Aug 12 '21 at 03:10
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    @Spectre The issue is not that the question is "simple" but rather that it is a dupe of a FAQ, with hundreds of prior good answers that already say all that can be said. Adding further (dupe) answers is strongly discouraged (unless they contain significant novelty - in which case they should be added to the dupe target). – Bill Dubuque Aug 12 '21 at 03:34
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    But more likely the dv is due to the fact that the above is far from a proof of existence and uniqueness of prime factorizations. – Bill Dubuque Aug 12 '21 at 04:01
  • @BillDubuque yeah, I saw that later. Actually, it was mistake that I didn't check for dupes. – Spectre Aug 12 '21 at 05:25
  • @MikeDesgrottes oh, don't worry. It already had an answer so it had to be closed. – Spectre Aug 12 '21 at 05:25