# What is this integer, associated to a list of rational numbers, actually called?

Suppose we're given a $k$-long list of rational numbers:

$$q : \{0,\ldots,k-1\} \rightarrow \mathbb{Q}.$$

Then there's a least $n \in \mathbb{Z}_{>0}$ such that $nq_i$ is an integer for all $i$.

Question. What is this integer $n$ called, and does it have any generalizations/related concepts in ring theory?

For what it's worth, I think $n$ can be computed as follows. Given a rational number $q$, lets define that the denominator of $q$ is the least $b \in \mathbb{Z}_{>0}$ such that $q=a/b$ for some integer $a$. This defines a function $$\mathrm{den}:\mathbb{Q} \rightarrow \mathbb{Z}_{>0}.$$ If I'm not mistaken, $n$ can be computed as $$\mathop{\mathrm{lcm}}_{i=0}^{k-1} \mathrm{den}(q_i).$$

Here's a "real" world example of how this shows up. Suppose we have a box with three kinds of chocolate; dark, white, and milk. We're given that if we reach into this box to pull out a random chocolate, there's $1/3$ probability of getting dark, $1/4$ probability of getting white, and hence $5/12$ probability of getting milk. The question is; what's the minimum number of chocolates in this box? The answer can be found by computing $n$ for the aforementioned rational numbers; this gives $12$, so that's the minimum number of chocolates that could be in the box.

• I don't think it has a name, but your formula to find it looks like it ought to be correct. – Arthur Jul 24 '17 at 13:03
• LCD – Daniel Fischer Jul 24 '17 at 13:08
• @DanielFischer, oh, haha. That makes so much sense. – goblin Jul 24 '17 at 13:09
• @DanielFischer As trivial as it is, you should post it as an answer. – orlp Jul 24 '17 at 13:43
• It seems that one of my serial downvoters decided to attack this answer. In case something is not clear there, please let me know and I will be happy to elaborate. – Bill Dubuque Jul 24 '17 at 22:00

$$n\,$$ is simply the least common denominator of the rationals $$\,q_i.\,$$ As you remarked, it is the lcm of the denominators when the fractions are written in least terms, since
$$n\, a_i/b_i\in\Bbb Z\iff b_i\mid n a_i\overset{(a_i,b_i)\,=\,1}\iff b_i\mid n\iff {\rm lcm}\{b_i\}\mid n$$