Why $[3,-1]$ is $-3$?

Question

According to the definition of LCM on Wikipedia,

LCM of $a$ and $b$ is the smallest positive integer that is divisible by both $a$ and $b$.

So, $[3,-1]$ should be $3$ but my book mentions it to be $-3$. Can you kindly clear the air?

Answer 1

Your book just has a different definition of LCM.

Everyone agrees what "least common multiple" means when we are talking about positive integers.

When we are talking about integers that may be negative, or more exotic numbers like $1+7i$, people may have different definitions that make sense, basically refer to the same concept, but are not exactly the same.

For the record, I personally think of $-3$ and $3$ as both being LCMs for $3$ and $-1$. Over the integers, including negatives, LCM is only unique up to $\pm$. But if that bothers you, your definition can require the LCM to be positive. There is some matter of opinion in these things.

Answer 2

I agree with other answerers that probably your text has a typo. But some people want the formula

$$\mbox{lcm}(a,b) = \frac{ab}{\gcd(a,b)} $$

to work. If the $-3$ is an answer in the back of the book, likely the author hired a student to work the exercises and that student had a different definition of lcm.