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I am trying to explain the least common multiple as precisely and clearly as possible, and so far I have come to the following definition.

lcm: is the smallest of all common (positive) multiples of 2 or more numbers.

With this definition I intend to cover doubts such as "if zero is a multiple of all numbers, why don't we consider it for the LCM?, and why are negative integer multiples not considered in the LCM?"

Thank you very much for reading, I hope you have a good day, greetings from Mexico.

Bill Dubuque
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Ángel Gutirrez
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    This is the most common definition for naturals (or integers). What other definition are you comparing to that you believe is less "precise" or "clear"? Perhaps the universal def: $, a,b\mid m\iff {\rm lcm}(a,b)\mid m?\ \ $ – Bill Dubuque May 30 '21 at 21:54
  • Adding zero to the list conveys no information. Negative integers also add nothing, since the positive integer is already there. – herb steinberg May 30 '21 at 22:07
  • @BillDubuque Thank you for answering, for example in the definition given by my textbook, translated from Spanish to English “The least common multiple is the least of the common multiples of 2 or more numbers”. Wikipedia's definition: In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm (a, b), is the smallest positive integer that is divisible by both a and b. – Ángel Gutirrez May 30 '21 at 22:38
  • @BillDubuque I suppose that the solution to the question of why zero and negative numbers are not taken into consideration is implicit in both, but I was not able to see it. Also, in all the videos that I have seen on the subject, when talking about multiples, zero is mentioned in any of them. – Ángel Gutirrez May 30 '21 at 22:39
  • So your definition is the same as Wiki's, and also the same as your textbook (if "number" (or "multiple") is restricted to positive integers - as is often the case in elementary treatises on divisibility). So it is not clear what is the point of your question. Are you asking about the reason why we normalize lcms to be positive [or monic (lead coef = 1) for polynomials over fields]? e.g. see here. – Bill Dubuque May 30 '21 at 23:23
  • @BillDubuque All my doubts are resolved, I just wanted to see if the way I am defining it was correct, thank you very much for the help. – Ángel Gutirrez May 30 '21 at 23:41

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