Least Common Multiple with ZeroDate: 03/26/2003 at 12:03:20 From: Pops Subject: Least Common Multiple with zero I'm trying to find any reference about the least common multiple of two numbers when one (or both) is zero. Could you help me, please? Date: 03/26/2003 at 15:46:31 From: Doctor Rick Subject: Re: Least Common Multiple with zero Hi, Pops. No number except zero is a multiple of zero, because zero times anything is zero. The only multiple that, say, 0 and 5 have in common is 0. Thus, if the LCM of 0 and 5 exists at all, it must be 0. We do not count zero as a common multiple. If we did, then zero would be the least common multiple of any two numbers (unless we also counted negative multiples, in which case there would be no least common multiple of any two numbers). Either we make an exception in this case, so that the LCM of zero and any number is zero, or we make no exception, in which case the LCM of zero and any number does not exist. To me it makes more sense to say that the LCM is defined only for positive numbers. See the definition of LCM here: Least Common Multiple - Eric Weisstein, World of Mathematics http://mathworld.wolfram.com/LeastCommonMultiple.html It says, "The least common multiple of two numbers a and b is the smallest number m for which there exist POSITIVE integers n_a and n_b such that n_a*a = n_b*b = m." [Emphasis is mine.] If the LCM of 0 and 5 were 0, we'd have a = 0, b = 5, n_a = any number, and n_b = 0 - which is not a positive integer, so it fails this definition. Thus, while the definition does not explicitly say that the two numbers must be positive, this is implied by the definition. I have to ask: Why do you care? Is there a context in which you need the LCM of zero and another number? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 03/28/2003 at 03:34:33 From: Pops Subject: Least Common Multiple with zero I'm a computer science professor and I'm proposing to the students a program to obtain the LCM of two numbers. My aim is to explain the correct answers in all possible "legal" situations. Thank you very much for your response. Pops. |
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