For ease of notation, in this section ‘number’ refers to a counting number: $\,1,\, 2,\, 3,\, \ldots\,$
The prime factorization of a number reveals its ‘multiplicative building blocks’—
Jo Johansen uses the words ‘personality’ and ‘structure’ to refer to the prime factorization.
The terms ‘unrelated’, ‘livesin’, and ‘overlapping’ (below) are also credited to Jo.
I love the images that this language connotes!
Jo has given me permission to share her insights about how ‘numbers converse with each other due to their personalities’ which gives ‘an efficient set of eyes through which the least common multiple (of two numbers) can be found expeditiously’.
There are three possible ways that two numbers can ‘relate’ to each other:


Two numbers are relatively prime
(informally: ‘unrelated’ or ‘strangers’)
when they have no prime factor in common.
Examples:

$8\,$ and $\,9\,$ are relatively prime $8\,$ and $\,9\,$ are unrelated $8\,$ and $\,9\,$ are strangers 
When two numbers are relatively prime (unrelated, strangers), then their least common multiple is their product:
Relatively prime is an important mathematical concept, and hence deserves a formal definition:
DEFINITION
relatively prime
Equivalently: 
The ‘livesin’ relationship says that one of the structures lives entirely inside the other.
That is, one of the numbers contains all the prime factors of the other.
It's easy to recognize the livesin case, because the smaller number goes into the bigger one evenly.
Examples:

$6 = 2\cdot 3$ $6\,$ ‘lives in’ $\,18$ 
In the ‘livesin’ case, the least common multiple is the bigger number.
It's as if the larger number says: ‘Not to worry!
I've got you covered, so I'll take care of the least common multiple!’
In the overlapping case, the structures share information between them, but one does not live in the other.
How can you recognize this case?
Firstly, it's not livesin,
so the smaller number doesn't go into the bigger evenly.
However, there is at least one prime factor that goes into both numbers evenly.
Example: $6\,$ overlaps $\,8\,$ (see image at right) Note: $\,6\,$ doesn't go into $\,8\,$ evenly. However, $\,2\,$ goes into both $\,6\,$ and $\,8\,$ evenly. 
$8 = 2\cdot 2\cdot 2$ $6\,$ overlaps $\,8$ shared factor: $\,2\,$ nonshared factors: $\,2\,$, $\,2\,$, and $\,3\,$ 
For overlapping numbers, the
Efficient Algorithm
for Finding the Least Common Multiple
(in a later section) is best.
Follow the link for a thorough discussion.
A quick overview is given next (which might be all you need):
The least common multiple of two numbers is the product of their shared and nonshared factors.
A shared factor is a common factor—it goes into both numbers evenly.
A nonshared factor is what's left over after factoring out the common factor.
Here's what it looks like for $\,6\,$ and $\,8\,$:
$$
\begin{align}
6 &= \color{green}{2}\cdot \color{red}{3}\cr\cr
8 &= \color{green}{2}\cdot \color{red}{2\cdot 2}
\end{align}
$$



shared factor in green: $2$ nonshared factors in red: $3\cdot 2\cdot 2 = 3\cdot 4$ $$ \begin{align} \cssId{s110}{\text{lcm }} &\cssId{s111}{= (\color{green}{\text{shared}})(\color{red}{\text{nonshared}})}\cr &\cssId{s112}{= \color{green}{2}\cdot (\color{red}{3}\cdot \color{red}{4})}\cr &\cssId{s113}{= 24} \end{align} $$ 
shared factor on the left: $2$ nonshared factors on the top: $3\cdot 4$ $$ \begin{align} \cssId{s121}{\text{lcm }} &\cssId{s122}{= (\color{green}{\text{shared}})(\color{red}{\text{nonshared}})}\cr &\cssId{s123}{= \color{green}{2}\cdot (\color{red}{3}\cdot \color{red}{4})}\cr &\cssId{s124}{= 24} \end{align} $$ 
Here are more examples of recognizing overlapping numbers and finding their least common multiple:
$48$ overlaps $54$ Why?

$420$ overlaps $1575$ Why?

CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.However, you can check to see if your answer is correct. 
PROBLEM TYPES:
