I want to gain intuition for least common multiple problems through a specific word problem involving cycle synchronization I'm trying to teach myself mathematics starting from arithmetic, and I'm stuck on a problem from an old math textbook (Arithmetic for the Practical Man): 

When several airplane engines are heard at the same time the sound is loudest when the explosions in all occur together, causing what are called "beats" in the roar of the engines. The explosions per second in the 4 engines of a great bomber are, respectively 660, 735, 735, 770. How many seconds elapse between the beats in the engine roar?

I know intuitively that the problem involves finding the least common multiple of the three different numbers, but after that I'm stuck. It occurs to me that I don't have a good sense for the LCM of 660, 735, and 770 really means, and that particular lack of intuition is a stumbling block. I'd like to understand the problem though dimensional analysis, which may be necessary for proper problem understanding for all I know! I also feel like a visual approach would be helpful for me to really grasp what's going on. Regardless, my purpose here is to gain some intuition for problems involving LCM.
Thanks everyone! This is my first post on Math Stack Exchange, so if I've messed up the etiquette, let me know. 
 A: If you just list the multiples of $2$ and $3$, you see they line up at the multiples of their least common multiple, $6$.  That is what is happening.
$$2,4,6,8,10,12,14,16,18,20,\ldots \\
3,6,9,12,15,18,21,\ldots $$
In your problem, you want the least common multiple of the inverses of the numbers given.  If one engine fires $660$ times per second, it fires every $\frac 1{660}$ second.  You want the least common multiple of $\frac 1{660}, \frac 1{735}$, and $\frac 1{770}$, which is $\frac 15$.  If you have them all fire together at time $0$, the first engine will fire again at times $\frac 1{660}, \frac 2{660}, \frac 3{660} \ldots$  This list includes $\frac 15=\frac {132}{660}$.  Similarly, $\frac 15=\frac {147}{735}=\frac{154}{770}$ so they will all fire again together at that time.  We start the cycle again and they all fire at $\frac 25$ seconds and so on.
A: The engine firings repeat in cycles of $\,1/660,\,1/735, 1/770\,$ seconds, so they are in unison at their common multiples. Using this formula for lcm of fractions, and the Euclidean gcd algorithm
${\rm lcm}\!\left(\dfrac{1}a,\dfrac{1}b,\dfrac{1}c\right) = \dfrac{1}{\color{#c00}{\gcd(a,b,c)}},\, $ $ \begin{align}\\ \&\,\ \color{#c00}{\gcd(660^{\phantom{|^{|^|}}}\!\!\!\!,735,770)} & \,=\,5\gcd(132,147,154)\\ &\,=\,5\gcd(15,\ \ 147,\ \ \ \ 7)\,=\,\color{#c00}5\end{align}$
