Background
I come into a problem where I need to define the least common Odd multiple.
Say I have two integer $a,b \in \mathbb{N}$, I want to define $c$ such that $a|c$ and $b|c$ in an oddly number fashion, meaning $\exists i \in \mathbb{Z}, c = (2i+1)a$, and $\exists j \in \mathbb{Z}, c=(2j+1)b$.
I want to define the smallest $c \in \mathbb{N}$ that satisfie the above, i.e., $\exists i,j \in \mathbb{Z}$ such those are true.
Can it be converted to a normal LCM? Or I have to define something like odd multiple?
If $a$ and $b$ have the same number of powers of $2$ in their factorizations the normal $\operatorname{LCM}$ will satisfy this. If they do not, there is no $c$ that will satisfy your desires because you must multiply the one with fewer powers of $2$ by an even number to bring the number of powers of $2$ up to the other one. For example, given $a=6,b=10$ that each have one power of $2$, the normal $\operatorname{LCM}$ is $30$, which is an odd number times each. Given $a=12,b=10$ the normal $\operatorname{LCM}$ is $60$, which multiplies $10$ by $6$ to get two powers of $2$ to match $12$.