# Relevancy to GCD and LCM

The traffic lights at three different road crossings change after every 48 sec, 72 sec, and 108 sec respectively. If they all change simultaneously at 8:20:00hours, then at what time will they change again simultaneously?

okay seriously how would someone even guess we have to do this by LCM GCD method ?

• Well, the first one changes at every multiple of $48$, the second at every multiple of $72$, and the third at every multiple of $108$...thus you want the least (positive) number which is a multiple of all three...which is the definition of the LCM.
– lulu
Sep 2 '17 at 22:46
• Why least number ?
– nban
Sep 2 '17 at 22:48
• Well, I took the question to be asking for the first time (after 8:20) that they all changed simultaneously. Doesn't matter much...if you want all the times then you still need to find the LCM (and the answer will then be all multiples of the LCM).
– lulu
Sep 2 '17 at 22:50
• That is what I am not understanding, why the concept of LCM ?
– nban
Sep 2 '17 at 22:52
• What part of my comment did you not understand? Can you see where the lights change simultaneously precisely at the common multiples of $48,72,108$?
– lulu
Sep 2 '17 at 22:55

It seems you need some help in understanding how $\,\rm lcm\,$ plays a role in such problems. Let's count time in seconds, relative to the starting time 8:20 when they all changed. Then light $\#1$ changes every $48$ seconds, i.e. at times $48n = 48, 96,\ldots =$ all multiples of $48,\,$ which are listed in the first row in the table below.  Similarly for lights $\#2$ and $\#3$ in the $2$nd and $3$rd rows.

$$\begin{array}{|r|r|rrrrrrrr|} \hline \#1 & 48n & 48 \!&\! \!&\! 96 \!&\! \!&\! \color{#c00}{144} \!&\! 192 \!&\! \!&\! 240 \!&\! \color{#c00}{288} \!&\! \!&\! 336 \!&\! \!&\! 384 \!&\! \color{#0a0}{432} \!&\! \ldots\, \\ \hline \#2 & 72n & \!&\! 72 \!&\! \!&\! \!&\! \color{#c00}{144} \!&\! \!&\! \color{#c00}{216} \!&\! \!&\! \color{#c00}{288} \!&\! \!&\! \!&\! 360 \!&\! \!&\! \color{#0a0}{432} \!&\! \ldots\,\\ \hline \#3 & 108n \!&\! \!&\! \!&\! \!&\! 108 \!&\! \!&\! \!&\! \color{#c00}{216} \!&\! \!&\! \!&\! 324 \!&\! \!&\! \!&\! \!&\! \color{#0a0}{432} \!&\! \ldots\,\\ \hline \end{array}$$

The times when $2$ of $3$ lights change are shown in $\rm\color{#c00}{red},$ and $\rm\color{#0a0}{green}$ shows the time when all $3$ change. By construction, the number $\color{#0a0}{432}$ is a common multiple of $48, 72,108.\,$ Since no prior common multiple of all three occurs, it is their least common multiple $(\rm lcm).$

Recall, as I explained in your prior question, the $\rm lcm$ is characterized by the universal property

$${ 48,72,108\mid m\iff \overbrace{{\rm lcm}(48,72,108)}^{\large \color{#0a0}{432}}\mid m}$$

This says that the times $m$ when all $3$ change  (i.e. when $m$ is a common multiple of $48,72,108),\,$ are equivalent to the times when $m$ is a multiple of their $\,{\rm lcm} = \color{#0a0}{432}.\,$ Interpreted in terms of the above table, this is true because the displayed pattern repeats if we consider the times $\!\!\pmod{\!\color{#0a0}{432}}$

• That's a Nice graphic! Sep 3 '17 at 0:30

here's an expansion on lulu's writing:

1. 8:20:00 all three are changing
2. 8:20:48 light 1 changes 48 seconds
3. 8:21:12 light 2 changes 72 seconds
4. 8:21:36 light 1 changes 96 seconds
5. 8:21:48 light 3 changes 108 seconds
6. 8:22:24 light 1 and light 2 change 144 seconds
7. 8:23:12 light 1 changes 192 seconds
8. 8:23:36 light 2 and light 3 change 216 seconds
9. 8:24:00 light 1 changes 240 seconds
10. 8:24:48 light 1 and light 2 change 288 seconds
11. 8:25:24 light 3 changes 324 seconds
12. 8:25:36 light 1 changes 336 seconds
13. 8:26:00 light 2 changes 360 seconds
14. 8:26:24 light 1 changes 384 seconds
15. ??? all three change ??? seconds

this took me very much more time than before bill's first comment to work out by mind when at least with computer I can compute the final time fairly fast with LCM so maybe they'd think to use it because it took them so much time otherwise.

• It would probably help the OP if you showed the time in seconds relative to the starting time, so the common multiples can be seen more clearly. Sep 2 '17 at 23:37

"okay seriously how would someone even guess we have to do this by LCM GCD method ?"

If the first light blinks ever $48$ seconds then any time it blinks will be at a mmultiple of $48$. So the answer when the all blink will have to be a multiple of $48$ because those are the only times the first one blinks.

The second only blinks at a multiple of $72$ seconds. So the answer has to be a common multiple of both $48$ and $72$.

And so on. The answer has to be a common multiple of all $48$, $72$, and $108$. What is a common multiple of $48$, $72$, and $108$.

Well... notice the first light will blink at: $48, 96, 144, 192, 240, 288, 336, 384, 432, 480,......$

The second light at: $72, 144, 216, 288, 360, 432, 504, 576, 648, 720 ....$

The third at: $108, 216, 324, 432, 540, 648, 756, 864, 972, 1080 ....$

What number do they all have in common? Well.... I can see it is $432$.

Dang. That was tedious. Maybe there is an easier way to figure out what is the least common mulitple of $48, 72$ and $108$.