# Is there an concept of Least Common Odd Multiple?

## 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?

• You basically ignore the prime $2$. You can define the lcm using the prime factorization $$lcm (p_1^{a_1} \cdots p_k^{a_k}, p_1^{b_1}\cdots p_k^{b_k}) = p_1^{\max\{a_1,b_1\}}\cdots p_k^{\max\{a_k,b_k\}}$$ and yours do the same just without $p_1$ Commented Sep 20, 2018 at 22:00
• @Yanko This is genius! Commented Sep 20, 2018 at 22:04
• @RossMillikan Sorry. I have corrected them. Commented Sep 20, 2018 at 22:04
• you can define anything you want any way you want. But this can't be done unless $a$ and $b$ both have the same power of $2$ as a divisor. You can find the lowest common odd mulitple of the odd components of $a$ and $b$ if you want. Commented Sep 20, 2018 at 22:15

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 $$12$$.

• Thank you for pointing this out. I wasn't realizing this fact. Indeed, this is becoming more interesting than I previously thought. Commented Sep 20, 2018 at 22:24
• The comment by Yanko is the standard way to determine the LCM of two numbers. It is one way to explain what I said above Commented Sep 20, 2018 at 22:46