Proposition: if a and b are integers, p is prime and p|ab then p|a or p|b
I'm getting started on math proofs, I want to know how can I write a simple valid proof of this without using the Bézout's identity, and making it as easier as posible.
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For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Henrik supports the community Mar 31 '24 at 21:56
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use the unique prime factorization – Vincent Batens Mar 31 '24 at 22:01
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This question has been answered a few times, such as here https://math.stackexchange.com/questions/2668515/prove-if-p-is-prime-and-p-%E2%88%A3-ab-then-p-%E2%88%A3-a-or-p-%E2%88%A3-b. It is worth doing a quick search before posting questions, or, if your question is different, try to make it really clear how your question is different. Your question seems to be a simple statement of Euclid's lemma as it is currently written. – Red Five Mar 31 '24 at 22:02
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1Use search, this question certainly has an answer here. For example, https://math.stackexchange.com/questions/1162373/if-prime-p-mid-ab-then-p-mid-a-or-p-mid-b-euclids-lemma. Different answers using different techniques. – Aig Mar 31 '24 at 22:03
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Does this answer your question? Prove: if p is prime and p ∣ ab, then p ∣ a or p ∣ b. – Red Five Mar 31 '24 at 22:03
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@RedFive OP wanted the proof without Bezout’s lemma, so your link is not quite what fits here. – Aig Mar 31 '24 at 22:04
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Is the Bezout lemma the same as Euclid's lemma? – Red Five Mar 31 '24 at 22:05
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1@RedFive No. – Aig Mar 31 '24 at 22:09
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Thanks @Aig - I didn't think so, but it has been a while since I studied number theory in proper depth. – Red Five Mar 31 '24 at 22:11
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If you don't want to use Bezout's lemma, you can use an inductive argument. See here: https://en.wikipedia.org/wiki/Euclid%27s_lemma – Ekene E. Mar 31 '24 at 22:29