Talk:Coprime integers

message: From the text:
"...As specific examples, 14 and 25 are coprime, being commonly divisible only by 1, while 14 and 21 are not coprime, because they are both divisible by 7..."
Huh?!?! If you're trying to say that "14 and 25 are coprime, sharing no factors in common between them except 1," then for God's sake, please say that! I puzzled over that text uncomprehendingly for a good two minutes, trying to decipher the odd phrase commonly divisible, which I recall seeing nowhere else; it was only when I stopped ruminating on the first example long enough to take in the second, counterexample, that understanding finally came. Good grief! The Grand Rascal (talk) 08:31, 28 September 2020 (UTC)
replies: text: , I agree. "Commonly divisible" is not a common phrase for saying "having a common divisor". I have fixed it. D.Lazard (talk) 08:51, 28 September 2020 (UTC)

text: [a] By chance, I read that a few days ago and found it perfectly understandable. The "proposed" version is also ok. [b] commonly divisible seems to be used: see (https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-prime-factorization-prealg/v/common-divisibility-examples). LMSchmitt 09:33, 28 September 2020 (UTC)
replies: text: Indeed, that's because it's a completely standard English language construction -- there's nothing jargony or obscure at all about it. --JBL (talk) 11:32, 28 September 2020 (UTC)
replies: text: , it is grammatically correct, but semantically confusing. In mathematics, one must always take care when words have a mathematical meaning that differ from their common (usual) meaning. In the case of "common divisor", "common" refers to "common to two integers". In "commonly divisible", "commonly" is not used in its common meaning, but is used in place of "simultaneously divisible". So the formulation is confusing. By the way Khan Academy is not a reliable source for attesting a common use of "commonly divisible" D.Lazard (talk) 14:35, 28 September 2020 (UTC)
replies: text:  To be clear: this is just forum-y chatting, I am not trying to get anyone to change the article. The most common meaning of "common" is probably "widespread; public".  The second-most common meaning of "common" is "shared; joint".  Both of these can be turned into an adverb: "commonly" meaning "frequently; often; usually" and "commonly" meaning "in common; jointly".  The phrase "greatest common divisor" uses the second meaning of the adjective "common", and the phrase "commonly divisible" uses the corresponding adverbial form.  Anyone who understands the phrase "greatest common divisor" should, by applying standard rules of English, be able to convert it to the corresponding adverbial form, and vice-versa.  P.S. I feel like it is strangely common (at least, this is not the first time) for you, a PhD mathematician who writes English well but not fluently, to lecture me, a PhD mathematician who writes English fluently, on basic points of English grammar and its use in mathematics.  I promise that I will never attempt to lecture you on fine points of French usage or grammar! --JBL (talk) 12:51, 29 September 2020 (UTC)
thread_title:  Awkward wording (a.k.a., "Huh?!" ) 

message: The lead can be difficult for someone first coming to this concept, even though the concept is not complicated.  I suggest that a sentence like the one Dan Harkless suggested be used, but with changes to avoid objections about 'coprime' not being a noon.  Another clarification would be to expand on the example of 8 and 9 by naming the prime factors of each (2 and 3). 2600:6C67:1C00:5F7E:6DF1:CD52:7E5F:D359 (talk) 15:25, 9 July 2022 (UTC)
thread_title:  Clarity in the lead 

message: What is co prime number 103.66.81.7 (talk) 17:04, 10 July 2022 (UTC)
replies: text: The answer is in the article. D.Lazard (talk) 18:06, 10 July 2022 (UTC)
thread_title:  Maths 

message: When three different random integers are selected, there are five different cases of the three numbers having a common divisor or being coprime to one another:
1. All three integers have a common divisor greater than 1.
   Example: 4, 6, 8; 4, 6, and 8 are all divisible by 2.
2. Exactly one pair of the three integers have a common divisor greater than 1 and the other integer is relatively prime to both the other two integers.
   Example: 3, 4, 10; 4 and 10 are divisible by 2, 3 is relatively prime to both 4 and 10.
3. Exactly two pairs of the three integers have a common divisor greater than 1 and one pair of integers is relatively prime.
   Example: 4, 6, 9; 4 and 6 are both divisible by 2, 6 and 9 are both divisible by 3, and 4 and 9 are relatively prime.
4. All three pairs of the three integers have a common divisor greater than 1 but the greatest common divisor of all three numbers is 1.
   Example: 6, 10, 15; 6 and 10 are both divisible by 2, 6 and 15 are both divisible by 3, 10 and 15 are both divisible by 5, but the greatest common divisor of 6, 10, and 15 is 1.
5. No two integers have a common divisor greater than 1 (pairwise coprime).
   Example: 3, 4, 5; 3, 4, and 5 are all relatively prime to one another.
What is the probability that each of the five cases will occur when choosing three random integers? Ar Colorado (talk) 16:02, 2 December 2022 (UTC)
replies: text: This depends on the probability law that is chosen, and, except for the case 4 (setwise coprimality), it should be an exercise of probability theory to deduce the probability from the case of  the probability of pairwise coprimality. In any case, per WP:VERIFIABILITY, a reliable source is needed for mentioning the result in this article, D.Lazard (talk) 16:39, 2 December 2022 (UTC)
thread_title:  Cases for three integers