How to find the LCM of One Negative and one positive Integer The title pretty much explains my question. While studying theory of numbers I came across this problem. The way I did LCM in childhood gave me a negative result.Maybe the method I used is wrong.
But according to the book,
LCM(-8,20)= 40
If I use the formula LCM(a,b)= |a.b|/GCD(a,b), Then I get the right answer. But this involves finding out gcd first. Is there a direct way to solve this problem?
Thank you in advance.
 A: It is defined that: the least 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. So the result must always be positive. 
Direct way to solve :Ignore the negative signs. Calculate as if everything's positive.
Hope I answered your question :)
A: An alternative to using the $\operatorname{lcm}(a,b)=\frac{|a\cdot b|}{\gcd(a,b)}$ relationship, is to break the absolute value of the numbers into their prime factors, and then multiply the highest powers of each prime (lcm by prime factorization).
For example, $|-8|=2^3$, and $|20|=2^2\cdot 5$, and so $\operatorname{lcm}(-8,20)=2^3\cdot5$.
A: Note : The LCM is defined as the least common multiple of the numbers that is positive. Or else, the answer would be $- \infty$. This explains it.
A: LCM of $a$ and $b$ can be defined in any commutative ring by the following universal property:
$$a,b \mid c\quad\text{and}\quad a,b\mid d\implies c\mid d$$
It's easy to prove that in integral domains, it's unique up to a multiplication by a unit.
In $\mathbb Z$, the units are $1, -1$, so LCM is unique up to a sign, so it would be more correct to say $\text{lcm}(-8,20)=\{40,-40\}$, but I guess your convention is always to take the positive result.
