I want to gain intuition for least common multiple problems through a specific word problem involving cycle synchronization

Question

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.

Answer 1

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.

Answer 2

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}$

Answer 3

"Least common multiple" means the least number that is a multiple they all have in common. It literally is what it says it is.

Perhaps a simpler example would be he least common multiple $4, 6, 9$.

The multiples of $4$ are $4$ and $8$ (which is $2*4$) and $12$ (which is $3*4$) and $16$ (which is $4*4$) and so on.

And the multiples of $6$ are $6$ and $12$ and $18$ and $24$ and so on.

And the multiples of $9$ are $9$ and $18$ and $27$ and $36$ and so on.

Now some numbers are multiples of two of the numbers but not of all three, such as $12$ is a multiple of $6$ and $4$ but not of $9$. Or $18$ is a multiple of $9$ and $6$ but not $4$. Those numbers are not multiples that all three have in common.

And some numbers are multiples of one of the numbers but not the other two. And some numbers (such as $5$ or $34$ or $21$) are not multiples of any of the three.

But SOME numbers are multiples of all three. For example $72$ is a multiple of $4$ and $72$ is a multiple of $6$ and $72$ is a multiple of $9$. So $72$ is a multiple all three have in common. $72$ is a common multiple of all of them.

$108$ is also a common multiple and so is $36$. And so is $16516224$. But what is the first (positive) number that is a common multiple of all three? Well, that is the least (it is the smallest) common (it is a multiple of not just on or two of them but of all three) multiple (it can be written as something times $4$, something else times $6$ or as something times $9$). And the least common multiple of $4,6,9$ is $36$. $36$ is a common multiple because $36$ is $4*9$ and $6*6$ and $9*4$ and it is the least because no smaller number has the property.

....

Now I think the actual problem you have having is that engine 1 doesn't explode after $660$ seconds. It explodes $660$ times PER second. And same for the other engines.

Had the question been gopher one emerges from his hole every $660$ minutes (so he emerges times: $660, 1320,1980, ... $ etc or any time $660t$) and gopher two emerges from her hole every $735$ minutes (so she emerges any time $735s$) and gopher 3 emerges every $770$ minutes, how often will all three emerge at the same time, then the answer would be the least common multiple of $660, 735,$ and $770$.

(Because for gopher one to emerge it must be a multiple of $660$, but for 2 and 3 to emerge it must also be a multiple of $735$ and of $770$ so in must be a common multiple of $660, 735,$ and $770$ and the first time it happens will be the least common multiple).

But that is NOT what the question is. In this case Engine 1 explodes $660$ times PER second or Engine 1 explodes once every $\frac 1{660}$ seconds. And Engine 2 (and 3) explodes $735$ times PER second or once every $\frac 1{735}$ seconds. And Engine 4 explodes once every $\frac 1{770}$ seconds.

So we DON"T want to find the least common multiple of $660, 735, 770$. We want to find the least common multiple of $\frac 1{660}, \frac 1{735}, \frac 1{770}$.

Now your reaction should be, mine was, "Wait! Those aren't integers! You can't find the least common multiple of fractions!" But...

of course you can. The multiples of $\frac 1{660}$ are $\frac 2{660}=\frac 1{330}$, $\frac 3{660}= \frac 1{220}, \frac 4{660} = \frac 1{165}$, etc. And the multiple of $\frac 1{735}$ are $\frac 2{735}, \frac 3{735} = \frac 1{245}$ etc. and so on. So what is the least common multiple of those.

.....

SO how do we find that?

Well heres a hint:

Engine 1: knocks at $\frac 1{660}$ of a second and and $\frac 2{660}=\frac 1{330}$ and $\frac 3{660}=\frac1{220}$ second and at $\frac 4{660}=\frac 1{165}$ and at $\frac 5{660} = \frac 1{132}$ and $\frac 6{660}=\frac 1{110}$ and at $\frac {10}{660}=\frac 1{66}$ and at $\frac {11}{660}=\frac 1{60}$ and at $\frac {12}{660}= \frac 1{55}$ and so one. if $660 = k*D$ then engine one will knock at $\frac k{660} =\frac 1D$.

Engine 2: knocks at $\frac 1{735}$ and at $\frac 3{735}= \frac 1{245}$ etc. If $735 = m*D$ then engine two will knock at $\frac {m}{735}=\frac 1{D}$.

But notice if $660 = k*D$ and $735= m*D$ then BOTH engines 1 and $ will knock at $\frac 15$.

Indeed: $660 = 132*5$ and $735= 147*5$. And engine 1 will explode at $\frac {132}{660}= \frac 15$ seconds. ANd engine 2 will explode at $\frac {147}{735} = \frac 15$ seconds. So they both explode at $\frac 15$ seconds.

So $\frac 15$ is a common multiple of $\frac 1{660}$ and of $\frac 1{730}$.

Is it a common multiple of $\frac 1{770}$? Is it the least common multiple?

And heres a hint to those questions: DO you think greatest common factors might come into play?