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If p|ab and (a,p)= 1 ax+py = 1 ⇒ axb+pyb=b ∴ p|b and p|a by symmetry.. (case if (b,p)= 1)

Kayan Tchian
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  • I don't know if this answers my question, but i knew that let a natural and p a prime, (a,p) = 1 or p|a

    Proof: I don't know if this answers my question, but i knew that let a natural and p a prime, (a,p) = 1 or p|a

    Proof: suppose by absurd that gcd(a,p) = d > 1 we have d|a and d|p, but p is prime, therefore d = p or d = 1, we assume d >1, therefore d = p

    ∴ if d = p ⇒ p|a

    And we can aply this Lemma and both numbers, can't?

     – Kayan Tchian Jan 24 '20 at 01:34
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    Do not confuse the exclusive or with the inclusive or. If $p$ is prime and $p\mid ab$ then one of the following three things is possible, 1) $p\mid a$ and $p\nmid b$, 2) $p\nmid a$ and $p\mid b$, 3) $p\mid a$ and $p\mid b$. When writing mathematics, if we say "$P$ or $Q$", we mean that at least one of $P$ or $Q$ are true, possibly only one, possibly both. – JMoravitz Jan 24 '20 at 01:39
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    Of course it is possible that both $p\mid a$ and $p\mid b$. Take for example $3\mid 9\times 24$. It is simultaneously true that both $3\mid 9$ and that $3\mid 24$. If we ever want to talk about the exclusive or, then we explicitly say so. (Some authors prefer to use "either...or" phrases to mean the exclusive or, but other authors might use "either...or" phrases to mean inclusive or instead... so it is better to avoid the issue altogether and use "or...but not both" phrases instead) – JMoravitz Jan 24 '20 at 01:42
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    In day to day English, probably most uses of "or" intend the exclusive or. In mathematical English, we generally mean inclusive or. – Will Jagy Jan 24 '20 at 01:48
  • so it is either ...... or ? – Kayan Tchian Jan 24 '20 at 01:54
  • thanks, guys I've seen this in my book of number theory and I had questions, the statement was exactly like the question. – Kayan Tchian Jan 24 '20 at 01:55
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    Quite the opposite... Saying "$p\mid a$ or $p\mid b$" has exactly the same meaning as saying "$p\mid a$ or $p\mid b$ or both." The "or both" part is implied and is left out but is still very much there. – JMoravitz Jan 24 '20 at 01:56
  • If I want to make it explicit that only one of them can be true, how would I write? – Kayan Tchian Jan 24 '20 at 01:59
  • As alluded to earlier, some authors (but not all) will implicitly use "either $p\mid a$ or $p\mid b$" to mean "$p\mid a$ is true or $p\mid b$ is true but not both", however that is not universally used. There are authors who will use "either...or" to still mean the inclusive or. As a result it is better to avoid that. I would use "...or...but not both" if you mean that only one can be true – JMoravitz Jan 24 '20 at 02:01
  • See conjunctions - Does "either A or B" preclude "both A and B" on english.stackexchange. See also logic - Does "either" make an exclusive or. – JMoravitz Jan 24 '20 at 02:02
  • @WillJagy FYI, regarding the usual meaning of "or" in day to day English, you may find the English Language & Usage SE post Conjunction - "or" - meaning. Is it eliminate the previous/afterwards alternative interesting. – John Omielan Jan 24 '20 at 02:03
  • It could. e.g. $p= 3, a =6, b = 9.$ What is relevant though, is that for all $a,b$ such that $p|ab$ then if follows that $p$ divides at least one of $a,b.$ A counter example might be helpful. If $p$ is not prime (e.g. $p=6),$ we can find $a,b$ (e.g. $a = 4, b = 3)$ such that $p|ab$, but $p$ does not divide $a$ or $b.$ – user317176 Jan 24 '20 at 02:06
  • I really must stress the fact that language is not always used correctly or accurately. Take for instance "I could care less" – JMoravitz Jan 24 '20 at 02:07
  • @Rottenbacher how would you describe the use of the most common Portuguese word for "or" ? – Will Jagy Jan 24 '20 at 02:17
  • some of the Portuguese varaiations of English "or" or translate: ou, nem, ou, senão, ou seja, quando não – Will Jagy Jan 24 '20 at 02:26

1 Answers1

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It might be "and" but it is not always so.

Consider this example:

$2$ is prime, and $2$ divides $(4\times 15)$.

Now $2$ does divide $4$, but $2$ does not divide $15$.

The point is: a prime number does not have any factors beside itself and 1.   Thus if a prime number is a factor of a product of two natural numbers, then it must be a factor of at least one among them.

Graham Kemp
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