What is the LCM of $x,2x, 3x$? My understanding says the lcm should be $6x$.
 A: The technique I use for finding the LCM of 3 numbers is best shown with an example. Let's assum we want the LCM of $ 8, 20, 50$, the first thing we do is a prime factorisation of each number:
$$8 = 2 \times 2 \times 2$$
$$20 = 2 \times 2 \times 5$$
$$50 = 2 \times 5 \times 5.$$
Now we first look at what number all three of them have in common. All three of them contain a $2$ and so the first number we have is :
$$2.$$
We are now left with 
$$8 \rightarrow 2 \times 2$$
$$20 \rightarrow 2 \times 5$$
$$50 \rightarrow 5 \times 5.$$
As we now don't have the same number in all three of them, we look for a number we have in two of them. Both $8$ and $20$ have a $2$ and both $20$ and $50$ have a $5$. So now multiplying all the numbers we have taken out together gives us
$$2 \times 2 \times 5.$$
We are now left with
$$8 \rightarrow 2$$
$$20 \rightarrow 0$$
$$50 \rightarrow 5.$$
Now, as we don't have the same number in any two of them, we just take the remainig number and multiply that with the numbers we have already multiplied, giving us
$$\mathrm{LCM} = 2 \times 2 \times 5 \times 2 \times 5 = 200,$$
which is the answer.
We can use this technique for your question. As we don't know if $x$ is a prime number, let's assume it is. Breaking up all three numbers gives us
$$x = x$$
$$2x = 2 \times x$$
$$3x = 3 \times x.$$
Now, the first number we take out is $x$ as they all have that in common. We are then left with
$$x \rightarrow 0$$
$$2x \rightarrow 2$$
$$3x \rightarrow 3.$$
As we don't have the same number in any two of them now, we just take out the remaining numbers and so to find our LCM we have to multiply
$$x \times 2 \times 3 = 6x.$$
So you are correct.
A: Another easy way is to "factor" a term out if $2+$ terms have the same factor. Then multiply the factors with what's left.
Let me explain with an example:

Find the LCM of $2,5,14,60$.

We notice $2,14,60$ can factor out a $2$. Thus, we have $$2| 1,5,7,30\tag1$$
Again, we notice that $5,30$ have common factor $5$. Thus, we have$$2|1,5,7,30\\ 5|1,1,7,\hspace{2mm}6$$
Multiplying what's remaining, we get the LCM as $2\cdot 5\cdot 1\cdot 1\cdot 7\cdot 6=\boxed{420}$

The same thing can be used to find the LCM of $x,2x,3x$. We notice that we can factor out $x$. Doing so, we obtain$$x|1,2,3$$
Since we can't factor anything out, we multiply everything to obtain $x\cdot 1\cdot 2\cdot 3=\boxed{6x}$
