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I get $2$ answers for this question: $$(t-1)^2 \quad \textrm{and} \quad -(t-1)^2$$

Which one is correct ? Why ? Is it a must for the LCM to be positive? Im confused. Please help.

azif00
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S. Fuard
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  • Usually I would consider the LCM of two positive numbers to be the smallest positive multiple of the two; otherwise you can wander off to $-\infty$ – Maximilian Janisch Jul 08 '19 at 00:29
  • Im pretty sure that the answer should be positive. But can it be negative in this case ? Why ? – S. Fuard Jul 08 '19 at 00:32
  • $(1-t)^2 = \left((-1)(t-1)\right)^2 = (t-1)^2$. If the LCM can be negative, $-(t-1)^2, -2(t-1)^2, -3(t-1)^2$ can be all lowest common multiples, which doesn't make sense. – Toby Mak Jul 08 '19 at 00:34
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    What is $,t,$, an integer, polynomial or $\ldots$? – Bill Dubuque Jul 08 '19 at 00:35

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Adding on Maximillian Janisch's comment:

$(t-1)^2$ is correct, since by definition the LCM must be positive (Wikipedia).

Consider finding the LCM of $3$ and $5$. If we restrict the LCM to be positive, then $15$ is the smallest common multiple. However, if the LCM can be negative, $-15, -30, -45 \cdots$ can also be the LCM. The LCM for negative numbers is not well defined, so it has to be restricted to the positive numbers.

Toby Mak
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  • Hmmm.. Got it. But in certain cases like finding the lcm of (b-a), 2(a-b) and 4a^2(a-b)^2 the answer is -4a^2(a-b)^2. In this case it can be negative right as we cant find a positive answer ? – S. Fuard Jul 08 '19 at 00:38
  • No, the LCM for however many numbers you have is always positive. If you want an example for $3$ numbers, try $2$, $3$ and $5$ and see that there are infinitely many negative LCMs. – Toby Mak Jul 08 '19 at 00:40
  • Yeah that is in the case of numbers right. It should be positive for numbers. And for algebraic expressions if there is no positive answer it has to b negative right? – S. Fuard Jul 08 '19 at 00:42
  • See this question. Since in calculating the LCM, you multiply the prime factors together (which is always positive), the LCM is always positive. In other words, you can just discard the negative sign. – Toby Mak Jul 08 '19 at 00:45
  • Hmmm.. Ok ok. I got it. THANK YOU SOOOO MUCH!!! – S. Fuard Jul 08 '19 at 00:46
  • No problem! Glad to have helped. – Toby Mak Jul 08 '19 at 00:47